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CHAPTER 7 Powers, Roots, and Scientific Notation

7.6 Chapter Review

Review Exercises

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?

Review Exercise Answers

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

Practice Test

In the following exercises, simplify each expression.

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.

Practice Test Answers

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

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Introductory Algebra Copyright © 2021 by Izabela Mazur is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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