7.6 Chapter Review

Izabela Mazur

CHAPTER 7 Powers, Roots, and Scientific Notation

7.6 Chapter Review

Simplify Expressions with Exponents

In the following exercises, simplify.

1. ${17}^{1}$	2. ${10}^{4}$
3. ${\left(0.5\right)}^{3}$	4. ${\left(\frac{2}{9}\right)}^{2}$
5. $-{2}^{6}$	6. ${\left(-2\right)}^{6}$

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression.

7. ${p}^{15}\cdot{p}^{16}$	8. ${x}^{4}\cdot{x}^{3}$
9. $8\cdot{8}^{5}$	10. ${4}^{10}\cdot{4}^{6}$
11. ${y}^{c}\cdot{y}^{3}$	12. $n\cdot{n}^{2}\cdot{n}^{4}$

Simplify Expressions Using the Power Property for Exponents

In the following exercises, simplify each expression.

13. ${\left({5}^{3}\right)}^{2}$	14. ${\left({m}^{3}\right)}^{5}$
15. ${\left({3}^{r}\right)}^{s}$	16. ${\left({y}^{4}\right)}^{x}$

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

17. ${\left(-5y\right)}^{3}$	18 ${\left(4a\right)}^{2}$
19. ${\left(10xyz\right)}^{3}$	20. ${\left(2mn\right)}^{5}$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

21. ${\left(4{a}^{3}{b}^{2}\right)}^{3}$	22. ${\left({p}^{2}\right)}^{5}\cdot{\left({p}^{3}\right)}^{6}$
23. ${\left(2{q}^{3}\right)}^{4}{\left(3q\right)}^{2}$	24. ${\left(5x\right)}^{2}\left(7x\right)$
25. ${\left(\frac{2}{5}{m}^{2}n\right)}^{3}$	26. ${\left(\frac{1}{3}{x}^{2}\right)}^{2}{\left(\frac{1}{2}x\right)}^{3}$

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

27. $\frac{{10}^{25}}{{10}^{5}}$	28. $\frac{{u}^{24}}{{u}^{6}}$
29. $\frac{{v}^{12}}{{v}^{48}}$	30. $\frac{{3}^{4}}{{3}^{6}}$
31. $\frac{5}{{5}^{8}}$	32. $\frac{x}{{x}^{5}}$

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

33. ${x}^{0}$	34. ${75}^{0}$
35. $\left(-{12}^{0}\right)$ ${\left(-12\right)}^{0}$	36. $-{12}^{0}$
37. ${\left(25x\right)}^{0}$	38. $25{x}^{0}$
39. ${\left(19n\right)}^{0}-{\left(25m\right)}^{0}$	40. $19{n}^{0}-25{m}^{0}$

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

41. ${\left(\frac{m}{3}\right)}^{4}$	42. ${\left(\frac{2}{5}\right)}^{3}$
43. ${\left(\frac{x}{2y}\right)}^{6}$	44. ${\left(\frac{r}{s}\right)}^{8}$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

45. $\frac{{n}^{10}}{{\left({n}^{5}\right)}^{2}}$	46. $\frac{{\left({x}^{3}\right)}^{5}}{{x}^{9}}$
47. ${\left(\frac{{r}^{8}}{{r}^{3}}\right)}^{4}$	48. ${\left(\frac{{q}^{6}}{{q}^{8}}\right)}^{3}$
49. ${\left(\frac{3{x}^{4}}{2{y}^{2}}\right)}^{5}$	50. ${\left(\frac{{c}^{2}}{{d}^{5}}\right)}^{9}$
51. $\frac{{\left(3{n}^{2}\right)}^{4}{\left(-5{n}^{4}\right)}^{3}}{{\left(-2{n}^{5}\right)}^{2}}$	52. ${\left(\frac{{v}^{3}{v}^{9}}{{v}^{6}}\right)}^{4}$

Divide Monomials

In the following exercises, divide the monomials.

53. $\frac{64{a}^{5}{b}^{9}}{-16{a}^{10}{b}^{3}}$	54. $-65{y}^{14}$ ÷ $5{y}^{2}$
55. $\frac{\left(8{p}^{6}{q}^{2}\right)\left(9{p}^{3}{q}^{5}\right)}{16{p}^{8}{q}^{7}}$	56. $\frac{144{x}^{15}{y}^{8}{z}^{3}}{18{x}^{10}{y}^{2}{z}^{12}}$

Use the Definition of a Negative Exponent

In the following exercises, simplify.

57. ${\left(-5\right)}^{-3}$	58. ${9}^{-2}$
59. ${\left(6u\right)}^{-3}$	60. $3\cdot{4}^{-3}$
61. ${\left(\frac{3}{4}\right)}^{-2}$	62. ${\left(\frac{2}{5}\right)}^{-1}$

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

63. ${q}^{-6}\cdot{q}^{-5}$	64. ${p}^{-2}\cdot{p}^{8}$
65. ${\left({y}^{8}\right)}^{-1}$	66. $\left({c}^{-2}d\right)\left({c}^{-3}{d}^{-2}\right)$
67. $\frac{{a}^{8}}{{a}^{12}}$	68. ${\left({q}^{-4}\right)}^{-3}$
69. $\frac{{r}^{-2}}{{r}^{-3}}$	70. $\frac{{n}^{5}}{{n}^{-4}}$

Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

71. 0.00429	72. 8,500,000
73. In 2015, the population of the world was about 7,200,000,000 people.	74. The thickness of a dime is about 0.053 inches.

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

75. 1.5 × ${10}^{10}$	76. 3.8 × ${10}^{5}$
77. 5.5 × ${10}^{-1}$	78. 9.1 × ${10}^{-7}$

Multiply and Divide Using Scientific Notation

In the following exercises, multiply and write your answer in decimal form.

79. 3.5 × ${10}^{-2})$ (6.2 × ${10}^{-1})$

80. 2 × ${10}^{5})$ (4 × ${10}^{-3})$

In the following exercises, divide and write your answer in decimal form.

81. $\frac{9\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-5}}{3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}}$

82. $\frac{8\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}}{4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-1}}$

Simplify Expressions with Square Roots

In the following exercises, simplify.

83. $\sqrt{144}$

85. $-\sqrt{81}$

87. $\sqrt{-36}$

89. $\sqrt{64+225}$

84. $\sqrt{64}$

86. $-\sqrt{25}$

88. $\sqrt{-9}$

90. $\sqrt{64}+\sqrt{225}$

Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

91. $\sqrt{155}$

92. $\sqrt{28}$

Approximate Square Roots

In the following exercises, approximate each square root and round to two decimal places.

93. $\sqrt{57}$

94. $\sqrt{15}$

Simplify Variable Expressions with Square Roots

In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)

95. $\sqrt{64{b}^{2}}$

97. $\sqrt{225{m}^{2}{n}^{2}}$

99. $\sqrt{49{y}^{2}}$

101. $\sqrt{121{c}^{2}{d}^{2}}$

96. $\sqrt{{q}^{2}}$

98. $-\sqrt{121{a}^{2}}$

100. $-\sqrt{100{q}^{2}}$

102. $\sqrt{4{a}^{2}{b}^{2}}$

Use Square Roots in Applications

In the following exercises, solve. Round to one decimal place.

103.Landscaping Janet wants to plant a square flower garden in her yard. She has enough topsoil to cover an area of $30$ square feet. How long can a side of the flower garden be?

105. Accident investigation The skid marks of a car involved in an accident were $216$ feet. How fast had the car been going before applying the brakes?

104. Art Diego has $225$ square inch tiles. He wants to use them to make a square mosaic. How long can each side of the mosaic be?

106. Gravity A hiker dropped a granola bar from a lookout spot $576$ feet above a valley. How long did it take the granola bar to reach the valley floor?

License

Icon for the Creative Commons Attribution 4.0 International License

1. 17	3. 0.125
5. $-64$	7. ${p}^{31}$
9. ${8}^{6}$	11. ${y}^{c+3}$
13. ${5}^{6}$	15. ${3}^{rs}$
17. $-125{y}^{3}$	19. $1000{x}^{3}{y}^{3}{z}^{3}$
21. $64{a}^{9}{b}^{6}$	23. $48{q}^{14}$
25. $\frac{8}{125}{m}^{6}{n}^{3}$	27. ${10}^{20}$
29. $\frac{1}{{v}^{36}}$	31. $\frac{1}{{5}^{7}}$
33. 1	35. 1
37. 1	39. 0
41. $\frac{{m}^{4}}{81}$	43. $\frac{{x}^{6}}{64{y}^{6}}$
45. 1	47. ${r}^{20}$
49. $\frac{343{x}^{20}}{32{y}^{10}}$	51. $-\frac{10,125{n}^{10}}{4}$
53. $-\frac{4{b}^{6}}{{a}^{5}}$	55. $\frac{9p}{2}$
57. $-\frac{1}{125}$	59. $\frac{1}{216{u}^{3}}$
61. $\frac{16}{9}$	63. $\frac{1}{{q}^{11}}$
65. $\frac{1}{{y}^{8}}$	67. $\frac{1}{{a}^{4}}$
69. $r$	71. $4.29\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-3}$
73. $7.2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{9}$	75. $15,000,000,000$
77. $0.55$	79. $0.0217$
81. $0.0000003$	83. 12
85. −9	87. not a real number
89. 17	91. $12<\sqrt{155}<13$
93. 7.55	95. 8b
97. 15mn	99. 7y
101. 11cd	103. 5.5 feet
105. 72 mph

1. ${\left(-\frac{2}{5}\right)}^{3}$	2. $u\cdot{u}^{4}$
3. ${\left(4{a}^{3}{b}^{5}\right)}^{2}$	4. $\frac{{n}^{-2}}{{n}^{-10}}$
5. $\frac{{3}^{8}}{{3}^{10}}$	6. ${\left(\frac{{v}^{2}{v}^{6}}{{v}^{4}}\right)}^{2}$
7. ${\left(87{x}^{15}{y}^{3}{z}^{22}\right)}^{0}$	8. ${\left(\frac{{m}^{4}\cdot m}{{m}^{3}}\right)}^{6}$
9. $\frac{80{c}^{8}{d}^{2}}{16c{d}^{10}}$	10. ${5}^{-2}$
11. ${q}^{-4}\cdot{q}^{-5}$	12. ${\left(4m\right)}^{-3}$
13. $\frac{8.4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-3}}{4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}}$	14. $\left(3.4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{9}\right)\left(2.2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-5}\right)$
15. $\sqrt{81}$	16. $-\sqrt{49}$
17. $\sqrt{-16}$	18. $\sqrt{{b}^{2}}$
19. $-\sqrt{64{a}^{2}}$	20. $-\sqrt{144{q}^{2}}$
21. Convert 83,000,000 to scientific notation.	22. Convert $6.91\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-5}$ to decimal form.

1. $-\frac{8}{125}$	2. ${u}^{5}$
3. $16{a}^{6}{b}^{10}$	4. ${n}^{8}$
5. $\frac{1}{9}$	6. ${v}^{8}$
7. $1$	8. ${m}^{12}$
9. $\frac{5{c}^7}{{d}^{8}}$	10. $\frac{1}{25}$
11. $\frac{1}{q}$	12. $\frac{1}{64 {m}^{3}}$
13. $2.1\times {10}^{-6}$	14. $7.48\times {10}^{4}$
15. 9	16. -7
17. not a real number	18. $b$
19. $-8a$	20. $-12q$
21. $8.3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{7}$	22. 0.0000691

Review Exercises

Simplify Expressions with Exponents

Simplify Expressions Using the Product Property for Exponents

Simplify Expressions Using the Power Property for Exponents

Simplify Expressions Using the Product to a Power Property

Simplify Expressions by Applying Several Properties

Simplify Expressions Using the Quotient Property for Exponents

Simplify Expressions with Zero Exponents

Simplify Expressions Using the Quotient to a Power Property

Simplify Expressions by Applying Several Properties

Divide Monomials

Use the Definition of a Negative Exponent

Simplify Expressions with Integer Exponents

Convert from Decimal Notation to Scientific Notation

Convert Scientific Notation to Decimal Form

Multiply and Divide Using Scientific Notation

Simplify Expressions with Square Roots

Estimate Square Roots

Approximate Square Roots

Simplify Variable Expressions with Square Roots

Use Square Roots in Applications

Review Exercise Answers

Practice Test

Practice Test Answers

License

Share This Book