Answer Key – Chapter 3

Exercise 3.1

1. a. Base: 7; Exponent: 4; Power: [latex]7^4[/latex]
   b. Repeated multiplication: [latex]9 \times 9 \times 9 \times 9 \times 9[/latex]; Base: 9; Exponent: 5
   c. Repeated multiplication: [latex]3 \times 3 \times 3 \times 3[/latex]; Base [latex]3^4[/latex]
   d. Base [latex]\displaystyle{\frac{2}{5}}[/latex]; Exponent: 6; Power: [latex]\displaystyle{\left(\frac{2}{5}\right)^6}[/latex]
   e. Repeated multiplication: [latex]\displaystyle{\frac{5}{7} \times \frac{5}{7} \times \frac{5}{7} \times \frac{5}{7} \times \frac{5}{7}}[/latex]; Base: [latex]\displaystyle{\frac{5}{7}}[/latex]; Exponent: 5
   f. Repeated multiplication: [latex]\displaystyle{\frac{4}{7} \times \frac{4}{7} \times \frac{4}{7}}[/latex]; Power: [latex]\displaystyle{\left(\frac{4}{7}\right)^3}[/latex]
   g. Base: 1.15; Exponent: 4; Power: [latex](1.15)^4[/latex]
   h. Repeated multiplication: [latex]1.6 \times 1.6 \times 1.6[/latex]; Base: 1.6; Exponent: 3
   i. Repeated multiplication: [latex]1.25 \times 1.25 \times 1.25 \times 1.25 \times 1.25[/latex]; Power: [latex](1.25)^5[/latex]

3. [latex]4^9 = 262,144[/latex]

5. [latex]\displaystyle{\left(\frac{1}{2}\right)^7 = 0.01}[/latex]

7. [latex]\displaystyle{\left(\frac{5}{2}\right)^5 = 97.66}[/latex]

9. [latex](3.25)^6 = 1,178.42[/latex]

11. [latex]6^5 = 7,776[/latex]

13. [latex]\displaystyle{\left(\frac{2}{5}\right)^2 = 0.16}[/latex]

15. [latex](1.4)^3 = 2.74[/latex]

17. [latex]6^6 = 46,656[/latex]

19. [latex]\displaystyle{\left(\frac{2}{3}\right)^{12} = 0.01}[/latex]

21. [latex](2.5)^6 = 244.14[/latex]

23. [latex]2^{10}[/latex]

25. [latex]3^8[/latex]

27. [latex]3^6[/latex]

29. [latex]2^8[/latex]

31. [latex]10^6[/latex]

33. [latex]2^6[/latex]

35. [latex]150[/latex]

37. [latex]609[/latex]

39. [latex]240[/latex]

41. [latex]58[/latex]

43. [latex]368[/latex]

45. [latex]7[/latex]

47. [latex]609[/latex]

49. [latex]257[/latex]

51. [latex]8,000[/latex]

53. [latex]625[/latex]

55. [latex]13,168.72[/latex]

57. [latex]91[/latex]

59. [latex]96[/latex]

61. [latex]\displaystyle{1\frac{3}{8} = 1.375}[/latex]

63. [latex]5.41[/latex]

65. [latex]5,184[/latex]

67. [latex]2.07[/latex]

69. [latex]104,976[/latex]

Exercise 3.2

1. a. [latex]\sqrt{64} = 8[/latex]
   b. [latex]\displaystyle{\sqrt{\frac{25}{16}} = \frac{5}{4} = 1.25}[/latex]

3. a. [latex]\sqrt[3]{8} = 2[/latex]
   b. [latex]\displaystyle{\sqrt[3]{\frac{27}{64}} = \frac{3}{4} = 0.75}[/latex]

5. a. [latex](144)^{\frac{1}{2}} = 12[/latex]
   b. [latex](64)^{\frac{1}{3}} = 4[/latex]

7. a. [latex](2)^{\frac{6}{2}} = 8[/latex]
   b. [latex](40)^{\frac{1}{2}} = 6.32[/latex]

9. a. [latex](8)^{\frac{1}{2}} \times (12)^{\frac{1}{2}} = 9.80[/latex]
   b. [latex](7)^{\frac{1}{2}} \times (14)^{\frac{1}{2}} = 9.90[/latex]

11. a. [latex]\displaystyle{\left(\frac{25}{49}\right)}^{\frac{1}{2}} = 0.71[/latex]
    b. [latex]\displaystyle{\left(\frac{64}{9}\right)}^{\frac{1}{2}} = 2.67[/latex]

13. a. [latex](2)^{\frac{3}{4}} = 1.68[/latex]
    b. [latex](5)^{\frac{3}{2}} = 11.18[/latex]

15. a. [latex](5)^{\frac{5}{4}} = 7.48[/latex]
    b. [latex](3)^{\frac{103}{72}} = 4.81[/latex]

17. a. [latex](8)^{\frac{7}{5}} = 18.38[/latex]
    b. [latex](5)^{\frac{5}{6}} = 3.82[/latex]

19. a. [latex]8^2 = 64[/latex]
    b. [latex](3)^{\frac{2}{3}} = 2.08[/latex]

21. a. [latex](4)^{\frac{3}{7}} = 1.81[/latex]
    b. [latex](3)^{\frac{2}{3}} = 2.08[/latex]

23. a. [latex]12^2 = 144[/latex]
    b. [latex]7^2 = 49[/latex]

25. a. [latex]4.88[/latex]
    b. [latex]1[/latex]

27. a. [latex]10.07[/latex]
    b. [latex]0.66[/latex]

29. a. [latex]8.49[/latex]
    b. [latex]51.96[/latex]

31. a. [latex]\displaystyle{\frac{7}{6} = 1.17}[/latex]
    b. [latex]0.19[/latex]

33. a. [latex]6^{-\frac{1}{2}} = \displaystyle{\frac{1}{\sqrt{6}} = 0.41}[/latex]
    b. [latex]7^{\frac{2}{3}} = \sqrt[3]{7^2} = 3.66[/latex]

35. a. [latex]10^{-\frac{1}{5}} = \displaystyle{\sqrt[5]{\frac{1}{10}} = 0.63}[/latex]
    b. [latex]2^{-\frac{9}{7}} = \displaystyle{\sqrt[7]{\left(\frac{1}{2}\right)^9} = 0.41}[/latex]

37. a. [latex]6^{-\frac{4}{3}} = \displaystyle{\sqrt[3]{\left(\frac{1}{6}\right)^4} = 0.09}[/latex]
    b. [latex]7^{\frac{37}{24}} = \sqrt[24]{7^{37}} = 20.08[/latex]

39. a. [latex]5^{-\frac{8}{3}} = \displaystyle{\sqrt[3]{\left(\frac{1}{5}\right)^8} = 0.01}[/latex]
    b. [latex]6^3 = 216[/latex]

41. a. [latex]8^4 = 4,096[/latex]
    b. [latex]7^{-3} = 0.003[/latex]

43. a. [latex]0.67[/latex]
    b. [latex]0.83[/latex]

45. a. [latex]0.33[/latex]
    b. [latex]8[/latex]

47. a. [latex]0.91[/latex]
    b. [latex]2.25[/latex]

Review Exercises 3

1. 7

3. [latex](3^5)^{\frac{3}{5}} = 27[/latex]

5. a. 2^2 = 4
   b. 5^3 = 125

7. a. 3^3 = 27
   b. 6^1 = 6

9. a. 27
   b. 0.2

11. a. 400
    b. –10,000,000

13. a. [latex]\displaystyle{\frac{1}{5} = 0.2}[/latex]
    b. [latex]\displaystyle{\frac{1}{7} = 0.\overline{142857}}[/latex]
    c. [latex]\displaystyle{\frac{8}{9} = 0.\overline{8}}[/latex]

15. a. 49
    b. [latex]\displaystyle{\frac{5}{6} = 0.8\overline{3}}[/latex]
    c. [latex]\displaystyle{\frac{6}{5} = 1.2}[/latex]

17. a. 7,180.08
    b. 1,817.28

19. 164,593.54

21. –17,668.97

Self-Test Exercises 3

1. a. [latex]3^6[/latex]
   b. [latex]2^7[/latex]
   c. [latex]3^3[/latex]
   c. [latex]2^3[/latex]

2. a. [latex]3^{10}[/latex]
   b. [latex]10^9[/latex]

3. a. [latex]\displaystyle{\frac{3^2}{2^6} = 0.14}[/latex]
   b. [latex]2 \times 3^2 = 18[/latex]

4. a. 0.01
   b. 1

5. a. 0.84
   b. 5.80

6. a. 13.6
   b. 1

7. a. 32
   b. 5.80

8. a. 1,953,125
   b. 3

9. a. –26
   b. 42

10. a. –61
    b. –16