LESSON Exponents 4-1 - Weeblylfandres.weebly.com/uploads/2/5/8/0/25801520/chapter… · · 2014-11-18LESSON 4-4 CONTINUED Example 2 Write 0.00709 in scientific notation. 0.00709

Copyright © by Holt McDougal. 69 Holt McDougal MathematicsAll rights reserved.

Lesson ObjectivesEvaluate expressions with exponents

Vocabulary

exponential form (p. 162)

exponent (p. 162)

base (p. 162)

power (p. 162)

Additional Examples

Example 1

Write in exponential form.

A. 4 ! 4 ! 4 ! 4

4 ! 4 ! 4 ! 4 ! Identify how many times is a factor.

B. ("6) ! ("6) ! ("6)

("6) ! ("6) ! ("6) ! Identify how many times ("6) is used as

a .

Example 2

Simplify.

A. 35

35 ! Find the product of 3’s.

!

Exponents4-1

LESSON

Simplify.

B. (!3)5

(!3)5 " Find the product of

!3’s.

"

LESSON 4-1 CONTINUED

Example 3

Evaluate x(yx ! zy) " xy for x # 4, y # 2, and z # 3.

x(yx ! zy) # xy

4(24 ! 32) # 42 Substitute for x, for y, and for z.

" 4( ! ) # Evaluate the .

" 4( ) # 16 inside the parentheses.

" # 16 from left to right.

" from left to right.

Example 4

Use the expression $12$(n2 ! 3n) to find the number of diagonals in a 7-sided

figure.

$12$(n2 ! 3n)

$12$(72 ! 3 ! 7) Substitute the number of sides for n.

$12$( ! ) Simplify inside the .

$12$( ) inside the parentheses.

.




Check It Out!

1. Write in exponential form.

7 ⋅ 7 ⋅ b ⋅ b

2. Simplify.

!(52)

3. Evaluate z ! 7(2x ! xy) for x " 5, y " 2, and z " 60.

4. Use the expression !12!(n2 " 3n) to find the number of diagonals in a

4-sided figure.


Integer Exponents4-2

LESSON

Lesson ObjectivesSimplify expressions with negative exponents and evaluate the zero exponent

Additional Examples

Example 1

Simplify. Write in decimal form.

A. 10!2

10!2 " #101! 10#

10!2 " "

B. 10!1

10!1 " #110#

10!1 " "

Example 2

A. Simplify 5!3.

!513! Write the ; change the of

the exponent.

#15# ! #

15# ! #

15# Find the product of #

15#’s.

B. Simplify (!10)!3.

" !#!110#"3 Write the ; change the of

the exponent.

" #!

110# ! #

!110# ! #

!110# Find the product of three ’s.

" # " !0.001 Simplify.



Example 3

Simplify 5 ! (6 ! 4)!3 " (!2)0.

5 ! (6 ! 4)!3 " (!2)0

# 5 ! ( )!3 " (!2)0 Subtract inside the .

# 5 ! ( ) " 1 Evaluate the .

# Add and subtract from left to right.

Check It Out!

1. Simplify. Write in decimal form.

10!8

2. Simplify (!4)!6.

3. Simplify 10 " (5 " 3)!2 " 50.


Properties of Exponents4-3

LESSON

Lesson ObjectivesApply the properties of exponents

Additional Examples

Example 1

Multiply. Write the product as one power.

A. 66 ! 63

exponents.

C. 25 ! 2

exponents.

Example 2

Divide. Write the quotient as one power.

A. !77

53!

exponents.

B. !xx190

!

Subtract .

Think: x1 !

B. n5 ! n7

exponents.

D. 244 ! 244

exponents.



Example 3

Simplify.

A. (54)2

exponents.

C. !!!23!"12""3

exponents.

Check It Out!

1. Multiply. Write the product as one power.

42 ! 44

2. Divide. Write the quotient as one power.

!99

9

2!

3. Simplify.

(5"2)"3

B. (67)9

exponents.

D. (172)"20

exponents.


Scientific Notation4-4

LESSON

Lesson ObjectivesExpress large and small numbers in scientific notation and compare two numberswritten in scientific notation

Vocabulary

scientific notation (p. 174)

Additional Examples

Example 1

Write each number in standard notation.

A. 1.35 ! 105

1.35 ! 105

1.35 ! 105 "

Think: Move the decimal right places.

B. 2.7 ! 10#3

2.7 ! 10#3

2.7 ! 10#3 "

2.7 1000 by the reciprocal.

Think: Move the decimal 3 places.

C. #2.01 ! 104

#2.01 ! 104

#2.01 ! 104 "

Think: Move the decimal right places.



Example 2

Write 0.00709 in scientific notation.

0.00709

Move the decimal to get a number between and .

7.09 ! 10 Set up notation.

Think: The decimal needs to move left to change 7.09 to

0.00709, so the exponent will be .

Think: The decimal needs to move places.

So 0.00709 written in scientific notation is ! .

Check ! " 7.09 ! 0.001 " 0.00709

Example 3

A pencil is 18.7 cm long. If you were to lay 10,000 pencils end-to-end, howmany millimeters long would they be? Write the answer in scientificnotation.

1 cm " mm

18.7 cm " mm Multiply each side by .

187 mm ! 10,000 Find the total length.

.

Set up scientific notation.

Think: The decimal needs to move to change 1.87 to 1,870,000, so the exponent will

be .

Think: The decimal needs to move places.


Example 4

A certain cell has a diameter of approximately 4.11 ! 10"5 meters.A second cell has a diameter of 1.5 ! 10"5 meters. Which cell has agreater diameter?

4.11 ! 10"5 1.5 ! 10"5

10"5 # 10"5 Compare powers of .

4.11 1.5 Compare the values between 1 and 10.

4.11 ! 10"5 1.5 ! 10"5

The has a greater diameter.

Check It Out!

1. Write the number in standard notation.

1.9 ! 10"5

2. Write 0.000811 in scientific notation.

3. An oil rig can hoist 2,400,000 pounds with its main derrick. It distributesthe weight evenly between 8 wire cables. What is the weight that eachwire cable can hold? Write the answer in scientific notation.

4. A certain cell has a diameter of approximately 5 ! 10"3 meters. A secondcell has a diameter of 5.11 ! 10"3 meters. Which cell has a greaterdiameter?


Lesson ObjectivesFind square roots

Vocabulary

square root (p. 182)

principal square root (p. 182)

perfect square (p. 182)

Additional Examples

Example 1

Find the two square roots of each number.

A. 49

!49" ! is a square root, since 7 ! 7 ! .

"!49" ! is also a square root, since "7 ! ("7) ! .

B. 100


"!100" ! is also a square root, since "10 ! ("10) !

.

C. 225


"!225" ! is also a square root, since "15 ! ("15) !

.


Squares and Square Roots4-5

LESSON



Example 2

A square window has an area of 169 square inches. How wide is thewindow?

Find the square root of to find the length of the window. Use the

square root; a negative square root has no meaning.

! 169

So !169" ! .

The window is inches wide.

Example 3

Simplify each expression.

A. 3!36" " 7

3!36" " 7 ! 3( ) " 7 Evaluate the root.

! " 7 Multiply.

! Add.

B. #$#2156#

" #34#

!25"!16" " #

34# Rewrite #$#215

0#as !25"

!16" .

! " #34# Evaluate the roots.

! Add.

Check It Out!

1. Find the two square roots of the number.

144


Estimating Square Roots4-6

LESSON

Lesson ObjectivesEstimate square roots and solve problems using square roots

Additional Examples

Example 1

Each square root is between two consecutive integers. Name theintegers. Explain your answer.

A. !55" Think: What are squares close to 55?

72 ! 49 " 55

82 ! 64 # 55

!55" is between and because is between and .

B. $!90" Think: What are perfect close to 90?

($9)2 ! 81 " 90

($10)2 ! 100 # 90

$!90" is between and because is between and

.



Guess 22.5

22.52 ! 506.25

Too high

Square root isbetween 22

and 22.5

Guess 22.2

22.22 ! 492.84

Too low

Square root isbetween 22.2

and 22.5

Guess 22.4

22.42 ! 501.76

Too high


and 22.4

Guess 22.3

22.32 ! 497.29

Too low


and 22.4

Example 2

You want to sew a fringe on a square tablecloth with an area of 500square inches. Calculate the length of each side of the tablecloth andthe length of fringe you will need to the nearest tenth of an inch.

Because 500 is between 222 and 232, the square root of 500 is between

and .

The square root is between 22.3 and 22.4. To round to the nearest tenth,look at the next decimal place. Consider 22.35.

22.352 ! Too

The square root must be greater than 22.35, so round up.

To the nearest tenth, !500" is about .

The length of each side of the tablecloth is about in.

The length of a side of the tablecloth is inches, to the nearest

tenth of an inch. Now estimate the length around the tablecloth.

! 22.4 ! Perimeter ! 4 ! side

You will need about inches of fringe.



Example 3

Estimate !141" to the nearest hundredth.

Step 1: Find the value of the whole number.

! 141 ! Find the perfect squares nearest 141.

! !141" ! Find the square roots of the perfect squares.

! !141" ! The number will be between and .

The whole number part of the answer is .

Step 2: Find the value of the decimal.

141 " 121 # Find the difference between the given

number, 141, and the lower perfect square.

144 " 121 # Find the difference between the greater

perfect square and the lower perfect square.

Write the difference as a ratio.

$ ≈ Divide to find the approximate decimal value.

Step 3: Find the approximate value.

% # Combine the whole number and decimal.

The approximate value of !141" to the nearest hundredth is .

Example 4

Use a calculator to find !600". Round to the nearest tenth.

Using a calculator, !600" # . . . Rounded, !600"

is .



Check It Out!

The square root is between two consecutive integers. Name the integers.

1. !80"

2. You want to build a fence around a square garden that is 250 squarefeet. Calculate the length of one side of the garden and the total lengthof the fence, to the nearest tenth.

3. Estimate !89" to the nearest hundredth.

4. Use a calculator to find !800". Round to the nearest tenth.



Lesson ObjectivesDetermine if a number is rational or irrational

Vocabulary

irrational number (p. 195)

real number (p. 195)

Density Property (p. 196)

Additional Examples

Example 1

Write all names that apply to each number.

A. !5" 5 is a number that is not a perfect .

B. !12.75 !12.75 is a decimal.

C. !16"2

!16"2 " #

42# " 2

The Real Numbers4-7

LESSON



Example 2

State if the number is rational, irrational, or not a real number. Justifyyour answer.

A. !03! !

03! " , because is a whole number

B. !#4" ; because it is the of a negative

number

C. #!49!$ !!

23!"!!

23!" " !

49! , !

23! is rational

Example 3

Find a real number between 3!25! and 3!

35!.

There are many solutions. One solution is halfway between the two numbers.To find it, add the numbers and divide by 2.

(3!25! $ 3!

35!) % 2 " !!!5 % 2 " % 2 " .

A real number between 3!25! and 3!

35! is .

Check It Out!

1. Write all names that apply to the number.

!9"

2. State if the number is rational, irrational, or not a real number.

!#7"""

3 43 51 3 5

2 3 53 3 5

4


Lesson ObjectivesUse the Pythagorean Theorem to solve problems

Vocabulary

Pythagorean Theorem (p. 200)

leg (p. 200)

hypotenuse (p. 200)

Additional Examples

Example 1

Find the length of each hypotenuse to the nearest hundredth.

A.

a2 ! b2 " c 2 Theorem2

!2

" c 2 Substitute for a and b.

! " c 2 Simplify powers.

" c 2

" c Solve for c ; c " !c2".

# c

5

c4

The Pythagorean Theorem4-8

LESSON



B. triangle with coordinates (1, !2), (1, 7), (13, !2)

The points form a right triangle with a ! 9 and b ! 12.

a2 " b2 ! c2 Pythagorean Theorem2

"2

! c2 Substitute for a and b.

" ! c2 Simplify .

! c2 .

! c Find the square root.

Example 2

Solve for the unknown side in the right triangle to the nearest tenth.

a2 " b2 ! c 2 Theorem2

" b2 ! Substitute for a and c.

" b2 ! Simplify powers.

# #

b2 !

b ! !576" ! 24

7

b25

x

y20

12

4

!4

!12

!20

!20 !12 !4 4 12 20



7

c5

4

b12

Example 3

Two airplanes leave the same airport at the same time. The first planeflies to a landing strip 350 miles south, while the other plane flies to anairport 725 miles west. How far apart are the two planes after they land?

a2 ! b2 " c2 Pythagorean Theorem2

!2

" c2 Substitute for a and b.

! " c2 Simplify .

" c2 .

! c Find the square root.

The planes are about 805 miles apart after they land.

Check It Out!

1. Find the length of the hypotenuse.

2. Solve for the unknown side in the right triangle.

3. Two birds leave the same spot at the same time. The first bird flies tohis nest 11 miles south, while the other bird flies to his nest 7 mileswest. How far apart are the two birds after they reach their nests?

Applying the Pythagorean Theoremand Its Converse4-9

LESSON


Lesson ObjectivesUse the Distance Formula and the Pythagorean Theorem and its converse tosolve problems

Additional Examples

Example 1

What is the diagonal length of the projector screen?

Find the length of the diagonal of the projector screen.

! " c2 Use the .

! " c2 Simplify.

" c2 Add.

" c Take the of both sides.

≈ c Find the .

The diagonal length is about feet.

7 ft

3 ft



xO 42

2

!2

!2

!4

!4

y

K M

LJ

Example 2

Find the distances between the points to the nearest tenth.

A. J and K

Let J be (x2, y2) and K be (x1, y1).

d ! !(x2 ""x1)2 #" (y2 "" y1)2" Use the .

! ! ( " )2

# ( " ( ))2

Substitute.

! ! ( )2

# 2

Subtract.

! ! # Simplify powers.

! ! ! Take the square root.

The distance between J and K is units.

Example 3

Tell whether the given side lengths form a right triangle.

A. 9, 12, 15

a2 # b2 c2 Compare a2 # b2 to c2.2

#2 2

Substitute.

# Simplify.

! ✓ Add.

The side lengths a right triangle.

Check It Out!1. A square garden has a side length of 10 meters. What is the length of the

diagonal of the garden, to the nearest hundredth?


4-1 Exponents

Simplify.

1. 43 2. 25 3. (!6)4 4. !73

Write in exponential form.

5. 10 6. 9 ! 9 ! 9 ! q ! q 7. !26 ! !26 ! !26 8. 5c ! 5c ! 5c ! 5c

4-2 Integer Exponents

Simplify. Write in decimal form.

9. 10!4 10. 100 11. 10!5 12. 10!3

Simplify.

13. (!2)!4 14. !3!2 15. 4!5 16. 10 ! 50 " (4 " 1)!2 ! (2 ! 1)!1

4-3 Properties of Exponents

Multiply. Write the product as one power.

17. 62 ! 65 18. y10 ! y10 19. 81 ! 80 20. a4 ! a!3

Divide. Write the quotient as one power.

21. #33

5

3# 22. #pp1

6

1# 23. #

r!r 7

14# 24. #cc

!

!

5

2#

Chapter Review4

CHAPTER


4-4 Scientific Notation

Write each number in standard notation.

25. 4.6 ! 108 26. "2.3 ! 10"3 27. 5.55 ! 1010

Write each number in scientific notation.

28. 7,456,000,000 29. "3,000,000,000,000 30. 0.000000204

31. The distance from Earth to Mars is 7,839,000 km. Suppose a rockettraveled from Earth to Mars and back 50 times. How many km did therocket travel? Write your answer in scientific notation.

4-5 Squares and Square Roots

Find the two square roots of each number.

32. 36 33. 625 34. 900 35. 289

Evaluate each expression.

36. !49" " 10 37. ##6146#$ 38. !49" ! !4"

4-6 Estimating Square Roots

Each square root is between two consecutive integers. Name theintegers.

39. !5" 40. "!43" 41. !1,000" 42. !75"

Use a calculator to find each value. Round to the nearest tenth.

43. !63" 44. !105" 45. !28.7" 46. !56"

CHAPTER 4 REVIEW CONTINUED


4-7 The Real Numbers

Write all names that apply to each number.

47. !81" 48. !11" 49. 3.75 50. !4

State if the number is rational, irrational, or not a real number.

51. !36" 52. #"1265"$ 53. "

50" 54. !13"

4-8 The Pythagorean Theorem

Solve for the unknown side of each right triangle to the nearest tenth.

55. 56. 57.

4-9 Applying the Pythagorean Theorem and Its Converse

Tell whether the given side lengths form a right triangle.

58. 7, 10, 12

58. 15, 20, 25

60. 9, 11, 14

61. A basketball court is 94 feet long and 50 feet wide. What is the length of a

diagonal of the basketball court, to the nearest tenth?

1115

a

60

b

36

12

c 16

CHAPTER 4 REVIEW CONTINUED


Answer these questions to summarize the important concepts fromChapter 4 in your own words.

1. Explain how to evaluate 36.

2. Explain the difference between 3.56 ! 108 and 3.56 ! 10"8.

3. Explain why !81" # $9.

4. Explain !"25" % $5.

5. Explain how to estimate !60".

6. Explain why 0.3" is a rational number.

For more review of Chapter 4:

• Complete the Chapter 4 Study Guide and Review on pages 212–214 ofyour textbook.

• Complete the Ready to Go On quizzes on pages 180 and 208 of yourtextbook.

Big Ideas4

CHAPTER

Documents

LESSON Exponents 4-1 - Weeblylfandres.weebly.com/uploads/2/5/8/0/25801520/chapter… · · 2014-11-18LESSON 4-4 CONTINUED Example 2 Write 0.00709 in scientific notation. 0.00709