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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. ⓐ d3·d645x·49x2y·4y3w·w2·w3

Answer

d9414x8y4w6

2. ⓐ x4·x289x·833z25·5z8y·y3·y5

3. ⓐ n19·n123x·367w5·8wa4·a3·a9

Answer

n313x+656w6
a16

4. ⓐ q27·q155x·54x9u41·7u53
c5·c11·c2

5. mx·m3

Answer

mx+3

6. ny·n2

7. ya·yb

Answer

ya+b

8. xp·xq

9. ⓐ x18x351253q18q36102103

Answer

x15591q18110

10. ⓐ y20y1071672t10t408385

11. ⓐ p21p741644bb9446

Answer

p144121b8145

12. ⓐ u24u391595xx710103

13. ⓐ 200b0

Answer

ⓐ 1 ⓑ 1

14. ⓐ 130k0

15. ⓐ 270(270)

Answer

11

16. ⓐ 150(150)

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

17. ⓐ a21031c5132

Answer

1a211000c59

18. ⓐ b41021c5152

19. ⓐ r31051q101103

Answer

1r31100,000q101,000

20. ⓐ s81021t91104

21. ⓐ (58)2(ba)2

Answer

6425a2b2

22. ⓐ (310)2(2z)3

23. ⓐ (49)3(uv)5

Answer

72964v5u5

24. ⓐ (72)3(3x)3

25. ⓐ (5)252(15)2(15)2

Answer

1251252525

26. ⓐ 53(15)3(15)3(5)3

27. ⓐ 3·51(3·5)1

Answer

35115

28. ⓐ 3·42(3·4)2

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

29. ⓐ b4b8(w4x5)(w2x4)) ⓒ (6c3d9)(2c4d5)

Answer

1b4w2x912cd4

30. ⓐ s3·s7(m3n3)(m5n1)
(2j5k8)(7j2k3)

31. ⓐ a3·a3(uv2)(u5v3)
(4r2s8)(9r4s3)

Answer

11u4v536r2j5

32. ⓐ y5·y5(pq4)(p6q3)
(5m4n6)(8m5n3)

33. p5·p2·p4

Answer

1p

34. x4·x2·x3

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

35. ⓐ (m4)2(103)6(x3)4

Answer

m810181x12

36. ⓐ (b2)7(38)2(k2)5

37. ⓐ (y3)x(5x)x(q6)8

Answer

y3x5xy1q48

38. ⓐ (x2)y(7a)b(a9)10

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

39. ⓐ (3xy)2(6a)0(5x2)2(4y3)2

Answer

9x2y2 ⓑ 1 ⓒ 125x416y6

40. ⓐ (4ab)2(5x)0(4y3)3(7y3)2

41. ⓐ (5ab)3(4pq)0(6x3)2(3y4)2

Answer

125a3b3 ⓑ 1 ⓒ 136x69y8

42. ⓐ (3xyz)4(7mn)0(3x3)2
(2y5)2

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

43. ⓐ (p2)5(xy)6(2xy2z)3(4p3q2)2

Answer

p532y6x68x3y6z3
16p6q4

44. ⓐ (x3)4(ab)5(2xy2z)3(x3yz4)2

45. ⓐ (a3b)4(54m)2(3a2b3c3)2(p1q4r4)2

Answer

a481b416m225a4c49b6q8r8p2

46. ⓐ (x2y)3(103q)4(2x3y43z2)5(5a3b12c4)3

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

47. ⓐ (5t2)3(3t)2(t2)5(t4)2(t3)7(2xy2x3y2)2(12xy3x3y1)1

Answer

1125t81t19y43x2

48. ⓐ (10k4)3(5k6)2(q3)6(q2)3(q4)8

49. ⓐ (m2n)2(2mn5)4(2p2)4(3p4)2(6p3)2

Answer

16m8n224p6

50. ⓐ (3pq4)2(6p6q)2(2k3)2(6k2)4(9k4)2

Mixed Practice

In the following exercises, simplify each expression.

51. ⓐ 7n1(7n)1(7n)1

Answer

7n17n17n

52. ⓐ 6r1(6r)1(6r)1

53. ⓐ (3p)23p23p2

Answer

19p23p23p2

54. ⓐ (2q)42q42q4

55. (x2)4·(x3)2

Answer

x14

56. (y4)3·(y5)2

57. (a2)6·(a3)8

Answer

a30

58. (b7)5·(b2)6

59. (2m6)3

Answer

2m18

60. (3y2)4

61. (10x2y)3

Answer

1,000x6y3

62. (2mn4)5

63. (2a3b2)4

Answer

16a12b8

64. (10u2v4)3

65. (23x2y)3

Answer

827x6y3

66. (79pq4)2

67. (8a3)2(2a)4

Answer

1,024a10

68. (5r2)3(3r)2

69. (10p4)3(5p6)2

Answer

25,000p24

70. (4x3)3(2x5)4

71. (12x2y3)4(4x5y3)2

Answer

x18y18

72. (13m3n2)4(9m8n3)2

73. (3m2n)2(2mn5)4

Answer

144m8n22

74. (2pq4)3(5p6q)2

75. ⓐ (3x)2(5x)(2y)3(6y)

Answer

45x348y4

76. ⓐ (12y2)3(23y)2(12j2)5(25j3)2

77. ⓐ (2r2)3(41r)2(3x3)3(31x5)4

Answer

12r413x11

78. (k2k8k3)2

79. (j2j5j4)3

Answer

1j3

80. (4m3)2(5m4)3(10m6)3

81. (10n2)3(4n5)2(2n8)2

Answer

4000n12

Use Scientific Notation

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

82. ⓐ 57,000 ⓑ 0.026

83. ⓐ 340,000 ⓑ 0.041

Answer

34×10441×103

84. ⓐ 8,750,000 ⓑ 0.00000871

85. ⓐ 1,290,000 ⓑ 0.00000103

Answer

1.29×106

103×108

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

86. ⓐ 5.2×1022.5×102

87. ⓐ 8.3×1023.8×102

Answer

830 ⓑ 0.038

88. ⓐ 7.5×1064.13×105

89. ⓐ 1.6×10108.43×106

Answer

ⓐ 16,000,000,000
ⓑ 0.00000843

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

90. ⓐ (3×105)(3×109)7×1031×107

91. ⓐ (2×102)(1×104)5×1021×1010

Answer

ⓐ 0.02 ⓑ 500,000,000

92. ⓐ (7.1×102)(2.4×104)6×1043×102

93. ⓐ (3.5×104)(1.6×102)8×1064×101

Answer

ⓐ 0.0000056 ⓑ 20,000,000

Writing Exercises

94. Use the Product Property for Exponents to explain why x·x=x2.

95. Jennifer thinks the quotient a24a6 simplifies to a4. What is wrong with her reasoning?

Answer

Answers will vary.

96. Explain why 53=(5)3 but 54(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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