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5.3E: Exercises

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Practice Makes Perfect

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

1. ⓐ d^3·d^64^{5x}·4^{9x}2y·4y^3w·w^2·w^3

Answer

d^94^{14x}8y^4w^6

2. ⓐ x^4·x^28^{9x}·8^33z^{25}·5z^8y·y^3·y^5

3. ⓐ n^{19}·n^{12}3^x·3^67w^5·8wa^4·a^3·a^9

Answer

n^{31}3^{x+6}56w^6
a^{16}

4. ⓐ q^{27}·q^{15}5^x·5^{4x}9u^{41}·7u^{53}
c^5·c^{11}·c^2

5. m^x·m^3

Answer

m^{x+3}

6. n^y·n^2

7. y^a·y^b

Answer

y^{a+b}

8. x^p·x^q

9. ⓐ \dfrac{x^{18}}{x^3}\dfrac{5^{12}}{5^3}\dfrac{q^{18}}{q^{36}}\dfrac{10^2}{10^3}

Answer

x^{15}5^9\dfrac{1}{q^{18}}\dfrac{1}{10}

10. ⓐ \dfrac{y^{20}}{y^{10}}\dfrac{7^{16}}{7^2}\dfrac{t^{10}}{t^{40}}\dfrac{8^3}{8^5}

11. ⓐ \dfrac{p^{21}}{p^7}\dfrac{4^{16}}{4^4}\dfrac{b}{b^9}\dfrac{4}{4^6}

Answer

p^{14}4^{12}\dfrac{1}{b^8}\dfrac{1}{4^5}

12. ⓐ \dfrac{u^{24}}{u^3}\dfrac{9^{15}}{9^5}\dfrac{x}{x^7}\dfrac{10}{10^3}

13. ⓐ 20^0b^0

Answer

ⓐ 1 ⓑ 1

14. ⓐ 13^0k^0

15. ⓐ −27^0−(27^0)

Answer

−1−1

16. ⓐ −15^0−(15^0)

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

17. ⓐ a^{−2}10^{−3}\dfrac{1}{c^{−5}}\dfrac{1}{3^{−2}}

Answer

\dfrac{1}{a^{2}}\dfrac{1}{1000}c^{5}9

18. ⓐ b^{−4}10^{−2}\dfrac{1}{c^{−5}}\dfrac{1}{5^{−2}}

19. ⓐ r^{−3}10^{−5}\dfrac{1}{q^{−10}}\dfrac{1}{10^{−3}}

Answer

\dfrac{1}{r3}\dfrac{1}{100,000}q^{10}1,000

20. ⓐ s^{−8}10^{−2}\dfrac{1}{t^{−9}}\dfrac{1}{10^{−4}}

21. ⓐ \left(\dfrac{5}{8}\right)^{-2}\left(−\dfrac{b}{a}\right)^{−2}

Answer

\dfrac{64}{25}\dfrac{a^{2}}{b^{2}}

22. ⓐ \left(\dfrac{3}{10}\right)^{−2}\left(−\dfrac{2}{z}\right)^{−3}

23. ⓐ \left(\dfrac{4}{9}\right)^{−3}\left(−\dfrac{u}{v}\right)^{−5}

Answer

\dfrac{729}{64}−\dfrac{v^{5}}{u^{5}}

24. ⓐ \left(\dfrac{7}{2}\right)^{−3}\left(−\dfrac{3}{x}\right)^{−3}

25. ⓐ (−5)^{−2}−5^{−2}\left(−\dfrac{1}{5}\right)^{−2}−\left(\dfrac{1}{5}\right)^{−2}

Answer

\dfrac{1}{25}−\dfrac{1}{25}25−25

26. ⓐ −5^{−3}\left(−\dfrac{1}{5}\right)^{−3}−\left(\dfrac{1}{5}\right)^{−3}(−5)^{−3}

27. ⓐ 3·5^{−1}(3·5)^{−1}

Answer

\dfrac{3}{5}\dfrac{1}{15}

28. ⓐ 3·4^{−2}(3·4)^{−2}

In the following exercises, simplify each expression using the Product Property.

29. ⓐ b^{4}b^{−8}(w^{4}x^{−5})(w^{−2}x^{−4})) ⓒ (−6c^{−3}d^9)(2c^4d^{−5})

Answer

\dfrac{1}{b^{4}}\dfrac{w^{2}}{x^{9}}−12cd^{4}

30. ⓐ s^{3}·s^{−7}(m^{3}n^{−3})(m^{5}n^{−1})
(−2j^{−5}k^{8})(7j^{2}k^{−3})

31. ⓐ a^{3}·a^{−3}(uv^{−2})(u^{−5}v^{−3})
(−4r^{−2}s^{−8})(9r^{4}s^{3})

Answer

1\dfrac{1}{u^{4}v^{5}}−36\dfrac{r^{2}}{j^{5}}

32. ⓐ y^{5}·y^{−5}(pq^{−4})(p^{−6}q^{−3})
(−5m^{4}n^{6})(8m^{−5}n^{−3})

33. p^{5}·p^{−2}·p^{−4}

Answer

\dfrac{1}{p}

34. x^{4}·x^{−2}·x^{−3}

In the following exercises, simplify each expression using the Power Property.

35. ⓐ (m^4)^2(10^3)^6(x^3)^{−4}

Answer

m^{8}10^{18}\dfrac{1}{x^{12}}

36. ⓐ (b^{2})^{7}(3^8)^2(k^2)^{−5}

37. ⓐ (y^3)^x(5^x)^x(q^6)^{−8}

Answer

y^{3x}5^{xy}\dfrac{1}{q^{48}}

38. ⓐ (x^2)^y(7^a)^b(a^9)^{−10}

In the following exercises, simplify each expression using the Product to a Power Property.

39. ⓐ (−3xy)^2(6a)^0(5x^2)^{−2}(−4y^{−3})^2

Answer

9x^2y^2 ⓑ 1 ⓒ \dfrac{1}{25x^4}\dfrac{16}{y^6}

40. ⓐ (−4ab)^2(5x)^0(4y^3)^{−3}(−7y^{−3})^2

41. ⓐ (−5ab)^3(−4pq)^0(−6x^3)^{−2}(3y^{−4})^2

Answer

−125a^3b^3 ⓑ 1 ⓒ \dfrac{1}{36x^6}\dfrac{9}{y^8}

42. ⓐ (−3xyz)^4(−7mn)^0(−3x^3)^{−2}
(2y^{−5})^2

In the following exercises, simplify each expression using the Quotient to a Power Property.

43. ⓐ (p^2)^5\left(\dfrac{x}{y}\right)^{−6}\left(\dfrac{2xy^2}{z}\right)^3\left(\dfrac{4p^{−3}}{q^2}\right)^2

Answer

\dfrac{p^5}{32}\dfrac{y^6}{x^6}\dfrac{8x^3y^6}{z^3}
\dfrac{16}{p^6q^4}

44. ⓐ \left(\dfrac{x}{3}\right)^4\left(\dfrac{a}{b}\right)^{−5}\left(\dfrac{2xy^2}{z}\right)^3\left(\dfrac{x^3y}{z^4}\right)^2

45. ⓐ \left(\dfrac{a}{3b}\right)^4\left(\dfrac{5}{4m}\right)^{−2}\left(\dfrac{3a^{−2}b^3}{c^3}\right)^{−2}\left(\dfrac{p^{−1}q^4}{r^{−4}}\right)^2

Answer

\dfrac{a^4}{81b^4}\dfrac{16m^2}{25}\dfrac{a^4c^4}{9b^6}\dfrac{q^8r^8}{p^2}

46. ⓐ \left(\dfrac{x^2}{y}\right)^3\left(\dfrac{10}{3q}\right)^{−4}\left(\dfrac{2x^3y^4}{3z^2}\right)^5\left(\dfrac{5a^3b^{−1}}{2c^4}\right)^{−3}

In the following exercises, simplify each expression by applying several properties.

47. ⓐ (5t^2)^3(3t)^2\dfrac{(t^2)^5(t^{−4})^2}{(t^3)^7}\left(\dfrac{2xy^2}{x^3y^{−2}}\right)^2\left(\dfrac{12xy^3}{x^3y^{−1}}\right)^{−1}

Answer

1125t^8\dfrac{1}{t^{19}}\dfrac{y^4}{3x^2}

48. ⓐ (10k^4)^3(5k^6)^2\dfrac{(q^3)^6(q^{−2})^3}{(q^4)^8}

49. ⓐ (m^2n)^2(2mn^5)^4\dfrac{(−2p^{−2})^4(3p^4)^2}{(−6p^3)^2}

Answer

16m^8n^{22}\dfrac{4}{p^6}

50. ⓐ (3pq^4)^2(6p^6q)^2\dfrac{(−2k^{−3})^2(6k^2)^4}{(9k^4)^2}

Mixed Practice

In the following exercises, simplify each expression.

51. ⓐ 7n^{−1}(7n)^{−1}(−7n)^{−1}

Answer

\dfrac{7}{n}\dfrac{1}{7n}−\dfrac{1}{7n}

52. ⓐ 6r^{−1}(6r)^{−1}(−6r)^{−1}

53. ⓐ (3p)^{−2}3p^{−2}−3p^{−2}

Answer

\dfrac{1}{9p^2}\dfrac{3}{p^2}−\dfrac{3}{p^2}

54. ⓐ (2q)^{−4}2q^{−4}−2q^{−4}

55. (x^2)^4·(x^3)^2

Answer

x^{14}

56. (y^4)^3·(y^5)^2

57. (a^2)^6·(a^3)^8

Answer

a^{30}

58. (b^7)^5·(b^2)^6

59. (2m^6)^3

Answer

2m^{18}

60. (3y^2)^4

61. (10x^2y)^3

Answer

1,000x^6y^3

62. (2mn^4)^5

63. (−2a^3b^2)^4

Answer

16a^{12}b^8

64. (−10u^2v^4)^3

65. \left(\dfrac{2}{3}x^2y\right)^3

Answer

\dfrac{8}{27}x^6y^3

66. \left(\dfrac{7}{9}pq^4\right)^2

67. (8a^3)^2(2a)^4

Answer

1,024a^{10}

68. (5r^2)^3(3r)^2

69. (10p^4)^3(5p^6)^2

Answer

25,000p^{24}

70. (4x^3)^3(2x^5)^4

71. \left(\dfrac{1}{2}x^2y^3\right)^4\left(4x^5y^3\right)^2

Answer

x^{18}y^{18}

72. \left(\dfrac{1}{3}m^3n^2\right)^4\left(9m^8n^3\right)^2

73. (3m^2n)^2(2mn^5)^4

Answer

144m^8n^{22}

74. (2pq^4)^3(5p^6q)^2

75. ⓐ (3x)^2(5x)(2y)^3(6y)

Answer

45x^348y^4

76. ⓐ \left(\dfrac{1}{2}y^2\right)^3\left(\dfrac{2}{3}y\right)^2\left(\dfrac{1}{2}j^2\right)^5\left(\dfrac{2}{5}j^3\right)^2

77. ⓐ (2r^{−2})^3(4^{−1}r)^2(3x^{−3})^3(3^{−1}x^5)^4

Answer

12r^413x^{11}

78. \left(\dfrac{k^{−2}k^8}{k^3}\right)^2

79. \left(\dfrac{j^{−2}j^5}{j^4}\right)^3

Answer

\dfrac{1}{j^3}

80. \dfrac{(−4m^{−3})^2(5m^4)^3}{(−10m^6)^3}

81. \dfrac{(−10n^{−2})^3(4n^5)^2}{(2n^8)^2}

Answer

−\dfrac{4000}{n^{12}}

Use Scientific Notation

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

82. ⓐ 57,000 ⓑ 0.026

83. ⓐ 340,000 ⓑ 0.041

Answer

34\times10^441\times10^{−3}

84. ⓐ 8,750,000 ⓑ 0.00000871

85. ⓐ 1,290,000 ⓑ 0.00000103

Answer

1.29\times10^6

103\times10^{−8}

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

86. ⓐ 5.2\times10^22.5\times10^{−2}

87. ⓐ −8.3\times10^23.8\times10^{−2}

Answer

−830 ⓑ 0.038

88. ⓐ 7.5\times10^6−4.13\times10^{−5}

89. ⓐ 1.6\times10^{10}8.43\times10^{−6}

Answer

ⓐ 16,000,000,000
ⓑ 0.00000843

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

90. ⓐ (3\times10^{−5})(3\times10^9)\dfrac{7\times10^{−3}}{1\times10^{−7}}

91. ⓐ (2\times10^2)(1\times10^{−4})\dfrac{5\times10^{−2}}{1\times10^{−10}}

Answer

ⓐ 0.02 ⓑ 500,000,000

92. ⓐ (7.1\times10^{−2})(2.4\times10^{−4})\dfrac{6\times10^4}{3\times10^{−2}}

93. ⓐ (3.5\times10^{−4})(1.6\times10^{−2})\dfrac{8\times10^6}{4\times10^{−1}}

Answer

ⓐ 0.0000056 ⓑ 20,000,000

Writing Exercises

94. Use the Product Property for Exponents to explain why x·x=x^2.

95. Jennifer thinks the quotient \dfrac{a^{24}}{a^6} simplifies to a^4. What is wrong with her reasoning?

Answer

Answers will vary.

96. Explain why −5^3=(−5)^3 but −5^4 \neq (−5)^4.

97. When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

Answer

Answers will vary.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “simplify expressions using the properties for exponents.”, “use the definition of a negative exponent”, and “use scientific notation”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

ⓑ After reviewing this checklist, what will you do to become confident for all goals?


This page titled 5.3E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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