Day 6 sept 15 lesson 1shy3 exponents continuednotebook

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Sep 12shy1122 AM May 13shy1026 AM

May 13shy1026 AM Sep 15shy907 AM

Day 6 sept 15 lesson 1shy3 exponents continuednotebook

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September 15 2015

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Day 6 sept 15 lesson 1shy3 exponents continuednotebook

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September 15 2015

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May 13shy1028 AM May 13shy1029 AM

May 13shy1029 AM May 13shy1029 AM

Day 6 sept 15 lesson 1shy3 exponents continuednotebook

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September 15 2015

May 13shy1029 AM May 13shy1029 AM

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Day 6 sept 15 lesson 1shy3 exponents continuednotebook

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Day 6 sept 15 lesson 1shy3 exponents continuednotebook

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Attachments

1shy3_Multiply_and_Divide_Monomialspdf

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 1

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 2

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 3

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 4

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 5

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 6

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 7

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 8

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 9

1-3 Multiply and Divide Monomials

Day 6 sept 15 lesson 1shy3 exponents continuednotebook

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Day 6 sept 15 lesson 1shy3 exponents continuednotebook

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September 15 2015

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Day 6 sept 15 lesson 1shy3 exponents continuednotebook

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September 15 2015

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Day 6 sept 15 lesson 1shy3 exponents continuednotebook

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Attachments

1shy3_Multiply_and_Divide_Monomialspdf

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 1

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 2

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 3

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 4

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 5

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 6

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 7

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 8

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 9

1-3 Multiply and Divide Monomials

Day 6 sept 15 lesson 1shy3 exponents continuednotebook

3

September 15 2015

Jun 19shy232 PM Sep 15shy908 AM

May 13shy1028 AM May 13shy1029 AM

May 13shy1029 AM May 13shy1029 AM

Day 6 sept 15 lesson 1shy3 exponents continuednotebook

4

September 15 2015

May 13shy1029 AM May 13shy1029 AM

May 13shy1029 AM Jun 19shy237 PM

May 13shy1030 AM Sep 15shy908 AM

Day 6 sept 15 lesson 1shy3 exponents continuednotebook

5

September 15 2015

Sep 15shy909 AM May 13shy1031 AM

May 13shy1031 AM Sep 15shy909 AM

May 13shy1031 AM May 13shy1031 AM

Day 6 sept 15 lesson 1shy3 exponents continuednotebook

6

September 15 2015

Sep 15shy1216 PM Jun 19shy233 PM

Attachments

1shy3_Multiply_and_Divide_Monomialspdf

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 1

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 2

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 3

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 4

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 5

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 6

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 7

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 8

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 9

1-3 Multiply and Divide Monomials

Day 6 sept 15 lesson 1shy3 exponents continuednotebook

4

September 15 2015

May 13shy1029 AM May 13shy1029 AM

May 13shy1029 AM Jun 19shy237 PM

May 13shy1030 AM Sep 15shy908 AM

Day 6 sept 15 lesson 1shy3 exponents continuednotebook

5

September 15 2015

Sep 15shy909 AM May 13shy1031 AM

May 13shy1031 AM Sep 15shy909 AM

May 13shy1031 AM May 13shy1031 AM

Day 6 sept 15 lesson 1shy3 exponents continuednotebook

6

September 15 2015

Sep 15shy1216 PM Jun 19shy233 PM

Attachments

1shy3_Multiply_and_Divide_Monomialspdf

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 1

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 2

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 3

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 4

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 5

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 6

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 7

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 8

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 9

1-3 Multiply and Divide Monomials

Day 6 sept 15 lesson 1shy3 exponents continuednotebook

5

September 15 2015

Sep 15shy909 AM May 13shy1031 AM

May 13shy1031 AM Sep 15shy909 AM

May 13shy1031 AM May 13shy1031 AM

Day 6 sept 15 lesson 1shy3 exponents continuednotebook

6

September 15 2015

Sep 15shy1216 PM Jun 19shy233 PM

Attachments

1shy3_Multiply_and_Divide_Monomialspdf

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 1

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 2

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 3

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 4

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 5

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 6

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 7

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 8

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 9

1-3 Multiply and Divide Monomials

Day 6 sept 15 lesson 1shy3 exponents continuednotebook

6

September 15 2015

Sep 15shy1216 PM Jun 19shy233 PM

Attachments

1shy3_Multiply_and_Divide_Monomialspdf

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 1

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 2

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 3

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 4

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 5

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 6

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 7

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 8

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 9

1-3 Multiply and Divide Monomials

Attachments

1shy3_Multiply_and_Divide_Monomialspdf

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 1

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 2

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 3

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 4

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 5

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 6

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 7

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 8

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 9

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 1

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 2

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 3

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 4

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 5

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 6

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 7

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 8

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 9

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 2

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 3

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 4

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 5

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 6

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 7

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 8

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 9

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 3

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 4

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 5

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 6

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 7

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 8

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 9

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 4

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 5

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 6

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 7

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 8

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 9

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 5

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 6

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 7

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 8

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 9

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 6

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 7

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 8

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 9

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 7

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 8

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 9

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 8

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 9

1-3 Multiply and Divide Monomials

Simplify using the Laws of Exponents

1 (ndash6)2 bull (ndash6)

5

SOLUTION The common base is ndash6 Add the exponents

2 ndash4a5(6a

5)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is a Add the exponents

3 (ndash7a4bc

3)(5ab

4c

2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common bases

are a b and c Remember that a = a1 and b = b

1

Add the exponents

4

SOLUTION The common base is 8 Find the difference between the exponents

5

SOLUTION Group the numbers and variables of the quotient The

common base is t Remember that t = t1 Find the

difference between the exponents Simplify

6

SOLUTION The common bases are x and y Find the difference between the exponents

7

SOLUTION Group the numbers and variables of the quotient The

common base is x Remember that 3 = 31 Find the

difference between exponents Simplify

8

SOLUTION Group the quotients by the common bases of 4 5

and 6 Remember that 6 = 61 Find the differences

between the exponents Simplify

9

SOLUTION The common base is 6 Work in the numerator separately from the denominator Add the exponentsFind the difference between the exponents Simplify

10

SOLUTION Group the quotients by the common bases of ndash2 ndash3

and ndash5 Remember that ndash3 = (ndash3)1 Find the

differences between the exponents Simplify

11 The processing speed of a certain computer is 1011

instructions per second Another computer has a

processing speed that is 103 times as fast How

many instructions per second can the faster computer process

SOLUTION The phrase times as fast indicates multiplication in this situation The common base is 10 Add the exponents

The faster computer can process 1014

instructions per second

12 The table shows the seating capacity of two differentfacilities About how many times as great is the capacity of Madison Square Garden in New York than a typical movie theater

SOLUTION

To find how many times as great divide 39 by 3

5

The common base is 3 Find the difference between the exponents

The capacity of Madison Square Garden is 34 or 81

times greater than a typical movie theater

13 Refer to the information in the table

a How many times as great is one quadrillion than one million b One quintillion is one trillion times as great as whatnumber

SOLUTION

a To find how many times as great divide 1015

by

106 The common base is 10 Find the difference

between the exponents

One quadrillion is 109 times greater than one million

Since 109 is also one billion one quadrillion is one

billion times greater than one million b To find how many times as great divide 10

18 by

1012

The common base is 10 Find the difference between the exponents

One quintillion is one trillion times as great as 106

Since 106 is also one million one quintillion is one

trillion times as great as one million

Persevere with Problems Find the missing exponent

14

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponents Let x represent the missing exponent x + 3 = 5 x = 2 So the missing exponent is 2

15

SOLUTION Since (3)(4) = 12 you need to find the exponent that

makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + 3 = 12 y = 9 So the missing exponent is 9

16

SOLUTION You need to find the exponent that makes

true To multiply powers with the same base add their exponentsLet x represent the missing exponent 3 + x + 2 = 9 5 + x = 9 x = 4 So the missing exponent is 4

17

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent x ndash 2 = 4 x = 6 So the missing exponent is 6

18

SOLUTION

You need to find the exponent that makes

true To divide powers with the same base subtract their exponents Let x represent the missing exponent 9 ndash x = 4 x = 5 So the missing exponent is 5

19

SOLUTION Simplify the left-hand side of the equation using the Commutative and Associative Properties and the Laws of Exponents

So You need to find the

exponent that makes true To multiply powers with the same base add their exponentsLet y represent the missing exponent y + (ndash4) = 3 y = 7 So the missing exponent is 7

20 Identify Structure Write a multiplication

expression with a product of 513

SOLUTION

Product indicates multiplication The exponents are added together if the bases of the powers being multiplied are the same So find two values whose sum is 13 and use them as exponents of 5 the

common base Sample answer 510

bull 53

21 Justify Conclusions Is greater than less

than or equal to 3 Explain your reasoning to a classmate

SOLUTION

is equal to 3 Sample answer Using the quotient

of powers = 3100 ndash 99

or 31 which is 3

22 Persevere with Problems What is twice 230

Write using exponents Explain your reasoning

SOLUTION

Twice is the same as 2 times Remember that 2 = 21

So twice 230

would be 2 times 230

23 Which expression is equivalent to 8x2y bull 8yz

2

A 64x2y

2z

2

B 64x2yz

2

C 16x2y

2z

2

D 384x2y

2z

2

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is y Remember that y = y1 Add the exponents

This corresponds with choice A

Simplify using the Laws of Exponents

24 (3x8)(5x)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base

is x Remember that x = x1 Add the exponents

25

SOLUTION The common base is h Find the difference between the exponents

26 2g2 bull 7g

6

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is g Add the exponents

27 (8w4)(ndashw

7)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is w Remember that ndashw is the same as ndash1w Add the exponents

28 (ndashp )(ndash9p2)

SOLUTION Use the Commutative and Associative Properties to group the numbers and variables The common base is p Remember that ndashp is the same as ndash1p and that

p = p1 Add the exponents

29

SOLUTION

The common base is 2 Remember that 2 = 21 Find

the difference between the exponents Simplify

30

SOLUTION Group the numbers and variables of the quotient Thecommon base is g Find the difference between the exponents Simplify

31

SOLUTION Group the quotients by the common bases of 5 and 7

Remember that 5 = 51 Find the differences between

the exponents Simplify

32

SOLUTION Group the quotients by the common bases of 4 and ndash

1 Remember that 4 = 41 Find the differences

between the exponents Simplify

33 Persevere with Problems The figure below is composed of a circle and a square The circle touches the square at the midpoints of the four sides

a What is the length of one side of the square

b The formula A = πr2 is used to find the area of a

circle The formula A = 4r2 can be used to find the

area of the square Write the ratio of the area of the circle to the area of the square in simplest form c Complete the table

d What can you conclude about the relationship between the areas of the circle and the square

SOLUTION a The length of one side of the square is the same asthe diameter of the circle The diameter of the circle is 2r so the length of one side of the square is 2r b Write the ratio of the area of the circle to the areaof the square Then simplify

c To complete the table find the area of each circlethe length of one side of the square that contains the circle and the area of the square Then write and simplify the ratio of the area of the circle to the area of the square Radius of 2 units You already know the areas of the circle and the square Write and simplify the ratio

Radius of 3 units Area of circle Substitute 3 for r in the formula A =

πr2 πr

2 = π(3)

2 or 9π units

2

Length of one side The length of one side of the square is 2r Substitute 3 for r in 2r 2r = 2(3) or 6 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(3)

2 or 36 units

2

Ratio

Radius of 4 units Area of circle Substitute 4 for r in the formula A =

πr2 πr

2 = π(4)

2 or 16π units

2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(4) or 8 units Area of square Substitute 3 for r in the formula A =

4r2 4r

2= 4(4)

2 or 64 units

2

Ratio

Radius of 2r units Area of circle Substitute 2r for r in the formula A =

πr2

πr2

= π(2r)2

= π(2r)(2r)

= 4πr2

units2

Length of one side The length of one side of the square is 2r Substitute 4 for r in 2r 2r = 2(2r) or 4r units Area of square Substitute 3 for r in the formula A =

4r2

4r2= 4(2r)

2

= 4(2r)(2r)

= 16r

2

Ratio

d The ratio of the area of the circle to the area of

the square will always be

34 One meter is 103 times longer than one millimeter

One kilometer is 106 times longer than one

millimeter How many times longer is one kilometer than one meter

A 109

B 106

C 103

D 10

SOLUTION

To find how many times longer divide 106 by 10

3

The common base is 10 Find the difference betweenthe exponents

One kilometer is 103 times longer than one meter

This corresponds to choice C

35 Which of the following is equivalent to

F

G

H

I

SOLUTION

Find the value of

The answer is G

36 Short Response What is the area of the rectangle below

SOLUTION The area of a rectangle is found by multiplying the length and the width Use the Commutative and Associative Properties to group the numbers and variables The common base is x Add the exponents

The area of the rectangle is 30x10

square feet

Multiply or divide37 14(ndash2)

SOLUTION The product of two integers with different signs is negative 14(ndash2) = ndash28

38 ndash20(ndash3)

SOLUTION The product of two integers with the same sign is positive ndash20(ndash3) = 60

39 ndash5(7)

SOLUTION The product of two integers with different signs is negative ndash5(7) = ndash35

40 ndash12 divide (ndash4)

SOLUTION The quotient of two integers with the same sign is positive ndash12 divide (ndash4) = 3

41 63 divide (ndash7)

SOLUTION The quotient of two integers with different signs is negative 63 divide (ndash7) = ndash9

42 250 divide (ndash50)

SOLUTION The quotient of two integers with different signs is negative 250 divide (ndash50) = ndash5

43 Three-fourths of a pan of lasagna is to be divided equally among 6 people What part of the lasagna will each person receive

SOLUTION

Divide by 6

So each person will receive of the lasagna

44 The tallest mountain in the United States is Mount

McKinley in Alaska The elevation is about 22 middot 5 middot

103 feet above sea level What is the height of Mount

McKinley

SOLUTION Evaluate the expression

22 middot 5 middot 10

3 = 4 middot 5 middot 1000

= 20000 The height of Mount McKinley is about 20000 feet above sea level

eSolutions Manual - Powered by Cognero Page 9

1-3 Multiply and Divide Monomials