Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

eSolutions Manual - Powered by Cognero Page 1

4-2 Negative Exponents

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

eSolutions Manual - Powered by Cognero Page 2

4-2 Negative Exponents

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

eSolutions Manual - Powered by Cognero Page 3

4-2 Negative Exponents

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

eSolutions Manual - Powered by Cognero Page 4

4-2 Negative Exponents

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

eSolutions Manual - Powered by Cognero Page 5

4-2 Negative Exponents

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

eSolutions Manual - Powered by Cognero Page 6

4-2 Negative Exponents

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

eSolutions Manual - Powered by Cognero Page 7

4-2 Negative Exponents

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

eSolutions Manual - Powered by Cognero Page 8

4-2 Negative Exponents

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

eSolutions Manual - Powered by Cognero Page 9

4-2 Negative Exponents

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

eSolutions Manual - Powered by Cognero Page 10

4-2 Negative Exponents

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

eSolutions Manual - Powered by Cognero Page 11

4-2 Negative Exponents

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

eSolutions Manual - Powered by Cognero Page 12

4-2 Negative Exponents

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

eSolutions Manual - Powered by Cognero Page 13

4-2 Negative Exponents

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

eSolutions Manual - Powered by Cognero Page 14

4-2 Negative Exponents

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

eSolutions Manual - Powered by Cognero Page 15

4-2 Negative Exponents

Write each expression using a positive exponent.

1. 

SOLUTION:  

=

ANSWER:  

2. 

SOLUTION:  

=

ANSWER:  

3. 

SOLUTION:  

=

ANSWER:  

4. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

5. 

SOLUTION:  

=

ANSWER:  

6. 

SOLUTION:  

=

ANSWER:  

7. 

SOLUTION:  

ANSWER:  

8. 

SOLUTION:  

ANSWER:  

9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 usinga negative exponent other than –1.

SOLUTION:  

ANSWER:  

Evaluate each expression if x = −4 and y = 2.

10. 

SOLUTION:  

ANSWER:  

11. 

SOLUTION:  

ANSWER:  

12. 

SOLUTION:  

ANSWER:  

13. 

SOLUTION:  

ANSWER:  

Write each expression using a positive exponent.

14. 

SOLUTION:  

=

ANSWER:  

15. 

SOLUTION:  

ANSWER:  

16. 

SOLUTION:  

=

ANSWER:  

17. 

SOLUTION:  

=

ANSWER:  

18. 

SOLUTION:  

=

ANSWER:  

19. 

SOLUTION:  

=

ANSWER:  

20. 

SOLUTION:  

=

ANSWER:  

21. 

SOLUTION:  

=

ANSWER:  

Write each fraction as an expression using a negative exponent other than −1.

22. 

SOLUTION:  

=

ANSWER:  

23. 

SOLUTION:  

=

ANSWER:  

24. 

SOLUTION:  

=

ANSWER:  

25. 

SOLUTION:  

=

ANSWER:  

26. 

SOLUTION:  

ANSWER:  

27. 

SOLUTION:  

ANSWER:  

28. 

SOLUTION:  

ANSWER:  

29. 

SOLUTION:  

ANSWER:  

Write each decimal using a negative exponent.30. The minimum thickness of Saturn’s A ring is one-

tenth kilometer.

SOLUTION:  

ANSWER:  

31. The diameter of a typical atom is 0.00000001 centimeter. Write the decimal using a negative exponent.

SOLUTION:  

ANSWER:  

Evaluate each expression if n = 3, p = −2, and q = 6.

32. 

SOLUTION:  

ANSWER:  

33. 

SOLUTION:  

ANSWER:  

34. 

SOLUTION:  

ANSWER:  

35. 

SOLUTION:  

ANSWER:  

36. 

SOLUTION:  

ANSWER:  

37. 

SOLUTION:  

ANSWER:  

38. 

SOLUTION:  

ANSWER:  

39. 

SOLUTION:  

ANSWER:  

40. 

SOLUTION:  

ANSWER:  

41. The table below shows the average lengths of different objects.

a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead?

c. A football field is about 102 meters long. How

many times as long is this than a cell?

SOLUTION:  a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom.

So, a virus is 103 or 1000 times as long as an atom.

b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus.

So, 104 or 10,000 viruses would fit across a pinhead.

c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell.

So, a football field is 106 or 1,000,000 times as long

as a cell.

ANSWER:  

a. 103 or 1000 times

b. 104 or 10,000

c. 106 or 1,000,000 times

42. The pH of a substance is a measure of its acidity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral. As the pH decreases, the substance is more acidic. The table shows the pH of several common substances.

a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentrationof one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water?

SOLUTION:  

a. Since  is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites.

So, the hydrogen ion concentration of coffee is 103 or

1000 times as great as that of egg whites.

b. One million = 1,000,000 or 106, so one millionth =

or 10−6

. Milk has a hydrogen ion

concentration of 10−6

or one millionth.

  c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water.

So, the hydrogen ion concentration of coffee is 102 or

100 times as great as that of pure water.

ANSWER:  

a. coffee, 103 or 1000 times

b. milk

c. 102 or 100 times

43. Be Precise A grain of sand has a volume of about

 cubic millimeter.

a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold

about 1010

grains of sand. What is the approximate volume of the sand art bottle?

c. If one cubic centimeter is equal to 103 cubic

millimeters, how many cubic centimeters of sand will the bottle hold?

SOLUTION:  a.

b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand.

So, the approximate volume of the sand art bottle is

106 mm

3.

c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter.

The bottle will hold 103 or 1000 cm

3 of sand.

ANSWER:  

a. 10−4

b. 106 mm

3

c. 103 cm

3

44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of ameter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent.    

SOLUTION:  

1 nanometer =  meter

To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer.

The greatest wavelength of an X-ray is 10−8

meters.

ANSWER:  

10−8

m

45. The shortest period of time ever measured directly was a light burst of a laser lasting about 0.000000000000001 second. Write this decimal as a fraction and as a power of ten.

SOLUTION:  

ANSWER:  

; 10−15

46. Multiple Representations In this problem, you willexplore negative exponents when using powers of 10,

10−1

= or 0.1.

a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents?Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the

value of 10−12

.

SOLUTION:  a.

b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

should have  or 12 zeros in the decimal

equivalent. 10−12

= 0.000000000001

ANSWER:  a.

b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For

example, 10−3

= 0.001.

d. 10−12

= 0.000000000001

47. Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the stepsyou would take to write the fraction as a power.

SOLUTION:  

ANSWER:  

48. Find the Error  Mahala is evaluating the expression

2 • 4−2

. Find her mistake and correct it.

SOLUTION:  Mahala added parentheses to the numbers in the expression. 

The correct solution is or .

ANSWER:  Mahala added parentheses to the numbers in the

expression. The correct solution is or .

49. Reason Inductively Consider the following sets of numbers:

Set 1: 2−2

, (−2)−2

, (−2)2, 2

2

Set 2: 2−3

, (−2)−3

, (−2)3, 2

3

a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list.

d. Finish the conjecture: 2−x

= (−2)−x

, if and only if ___________.

e . Finish the conjecture: (−2)x = 2

x , if and only if  

__________.

SOLUTION:  a.

22 = 4

2−2

= (−2)−2

and (−2)2 = 2

2

b.

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative.

d. 2−x = (−2)

−x , if and only if x is an even number.

e. (−2)x = 2

x , if and only if x is an even number.

ANSWER:  

a. , , , 22 = 4;

2−2

= (−2)−2

and (−2)2 = 2

2

b. , , , ;

None of the expressions are equal. c. Sample answer: When you square either a positiveor a negative value, the answer is positive. When youcube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number

50. Persevere with Problems Compare and contrast

x−n

and xn where x ≠ 0. Then give a numerical 

example to show the relationship.

SOLUTION:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

ANSWER:  

They are multiplicative inverses. Sample answer: 2−4

and 24 are multiplicative inverses because 2

−4 = ,

and .

51. Justify Conclusions Investigate the fraction .

Does it increase or decrease as the value of n increases? Explain.

SOLUTION:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

ANSWER:  

Sample answer: If n = 3,   If n = 4,

 So, as the value of n increases, the

value of decreases.

52. Building on the Essential Question Explain the

difference between the expressions (−3)4 and 3

−4.

SOLUTION:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

ANSWER:  

Sample answer: (−3)4 is the same as (−3)(−3)(−3)

(−3) or 81. 3−4

is the same as  or  .

53. DNA contains the genetic code of an organism. The

length of a DNA strand is about 10−7

meter. Which of the following represents the length of the DNA strand as a decimal?

 

A  0.00001 m

B  0.000001 m

C  0.0000001 m

D  0.00000001 m

SOLUTION:  

The length of the DNA strand is 0.0000001 meters. Choice C is correct.

ANSWER:  C

54. When simplified, 2−5

is equal to which of the following?

 

F  −32

G  −

H  

J  32

SOLUTION:  

Choice H is correct.

ANSWER:  H

55. Which of the following shows the expressions 40,

4−2

, 42, and 4

−3 in order from least to greatest?

 

A  4−3, 4

−2, 4

2, 4

0

B  40 , 4

−2, 4

−3, 4

2

C  42, 4

0, 4

−2, 4

−3

D  4−3, 4

−2, 4

0, 4

2

SOLUTION:  Since the bases are all equal, arrange the exponents from least to greatest.

−3 < −2 < 0 < 2, so 4−3

< 4−2

< 40 < 4

2

Choice D is correct.

ANSWER:  D

56. SHORT RESPONSE It takes light 5.3 × 0.000001 seconds to travel one mile. Write 0.000001 as a fraction and as a power of 10.

SOLUTION:  

ANSWER:  

; 10−6

57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest.

SOLUTION:  Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are –509, –505, –435, –410.

ANSWER:  –509, –505, –435, –410

Find each sum or difference. Write in simplest form.

58. 

SOLUTION:  

ANSWER:  

59. 

SOLUTION:  

ANSWER:  

60. 

SOLUTION:  

ANSWER:  

61. 

SOLUTION:  

ANSWER:  

62. 

SOLUTION:  

ANSWER:  

63. 

SOLUTION:  

ANSWER:  

64. Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months?

SOLUTION:  The integer –75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply –75 by 3. –75 × 3 = –225 A total withdrawal of $225 or –$225.

ANSWER:  a withdrawal of $225 or –$225

Evaluate each expression.65. 4n if n = –12

SOLUTION:  

ANSWER:  –48

66. –9p if p = –7

SOLUTION:  

ANSWER:  63

67. 3xy if x = –4 and y = 5

SOLUTION:  

ANSWER:  –60

68. –8st if s = –2 and t = 6

SOLUTION:  

ANSWER:  96

Write each expression using exponents.69. 15 • 15 • 15 • 15 • 15

SOLUTION:  The base 15 is a factor 5 times. So, the exponent is 5.

15 • 15 • 15 • 15 • 15 = 155

 

ANSWER:  

155

70. bc • bc • bc • bc • bc • bc • bc • bc

SOLUTION:  The base bc is a factor 8 times. So, the exponent is 8.

bc • bc • bc • bc • bc • bc • bc • bc = (bc)8

or b8c

8

 

ANSWER:  

(bc)8

or b8c

8

71. 4 • 9 • x • x • y • y • y • y

SOLUTION:  

ANSWER:  

36x2y

4

72. (p + 2)(p + 2)(p + 2)

SOLUTION:  The base (p + 2) is a factor 3 times. So, the exponentis 3.

(p + 2)(p + 2)(p + 2) = (p + 2)3

ANSWER:  

(p + 2)3

Find each product.73. 25 × 0.001

SOLUTION:  

ANSWER:  0.025

74. 107 × 0.0001

SOLUTION:  

ANSWER:  0.0107

75. 3.8 × 0.01

SOLUTION:  

ANSWER:  0.038

76. 0.5 × 0.021

SOLUTION:  

ANSWER:  0.0105

77. 1.5 × 0.003

SOLUTION:  

ANSWER:  0.0045

78. 4.2 × 0.0005

SOLUTION:  

ANSWER:  0.0021

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4-2 Negative Exponents