Chapter 3

Exercise 3.1

1. 1. Base: 7; Exponent: 4; Power: [latex] 7^4 [/latex]

      
      

   2. Repeated multiplication: [latex] 9 \times 9 \times 9 \times 9 \times 9 [/latex]; Base: 9; Exponent: 5

      
      

   3. Repeated multiplication: [latex] 3 \times 3 \times 3 \times 3 [/latex]; Base [latex] 3^4 [/latex]

      
      

   4. Base [latex] \displaystyle{\frac{2}{5}} [/latex]; Exponent: 6; Power: [latex] \displaystyle{\left(\frac{2}{5}\right)^6} [/latex]

      
      

   5. 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

      
      

   6. 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]

      
      

   7. Base: 1.15; Exponent: 4; Power: [latex] (1.15)^4 [/latex]

      
      

   8. Repeated multiplication: [latex] 1.6 \times 1.6 \times 1.6 [/latex]; Base: 1.6; Exponent: 3

      
      

   9. 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. 1. [latex] \sqrt{64} = 8 [/latex]

      
      

   2. [latex] \displaystyle{\sqrt{\frac{25}{16}} = \frac{5}{4} = 1.25} [/latex]

      
      

3. 1. [latex] \sqrt[3]{8} = 2 [/latex]

      
      

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

      
      

5. 1. [latex] (144)^{\frac{1}{2}} = 12 [/latex]

      
      

   2. [latex] (64)^{\frac{1}{3}} = 4 [/latex]

      
      

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

      
      

   2. [latex] (40)^{\frac{1}{2}} = 6.32 [/latex]

      
      

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

      
      

   2. [latex] (7)^{\frac{1}{2}} \times (14)^{\frac{1}{2}} = 9.90 [/latex]

      
      

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

       
       

    2. [latex] \displaystyle{\left(\frac{64}{9}\right)}^{\frac{1}{2}} = 2.67 [/latex]

       
       

13. 1. [latex] (2)^{\frac{3}{4}} = 1.68 [/latex]

       
       

    2. [latex] (5)^{\frac{3}{2}} = 11.18 [/latex]

       
       

15. 1. [latex] (5)^{\frac{5}{4}} = 7.48 [/latex]

       
       

    2. [latex] (3)^{\frac{103}{72}} = 4.81 [/latex]

       
       

17. 1. [latex] (8)^{\frac{7}{5}} = 18.38 [/latex]

       
       

    2. [latex] (5)^{\frac{5}{6}} = 3.82 [/latex]

       
       

19. 1. [latex] 8^2 = 64 [/latex]

       
       

    2. [latex] (3)^{\frac{2}{3}} = 2.08 [/latex]

       
       

21. 1. [latex] (4)^{\frac{3}{7}} = 1.81 [/latex]

       
       

    2. [latex] (3)^{\frac{2}{3}} = 2.08 [/latex]

       
       

23. 1. [latex] 12^2 = 144 [/latex]

       
       

    2. [latex] 7^2 = 49 [/latex]

       
       

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

       
       

    2. [latex] 1 [/latex]

       
       

27. 1. [latex] 10.07 [/latex]

       
       

    2. [latex] 0.66 [/latex]

       
       

29. 1. [latex] 8.49 [/latex]

       
       

    2. [latex] 51.96 [/latex]

       
       

31. 1. [latex] \displaystyle{\frac{7}{6} = 1.17} [/latex]

       
       

    2. [latex] 0.19 [/latex]

       
       

33. 1. [latex] 6^{-\frac{1}{2}} = \displaystyle{\frac{1}{\sqrt{6}} = 0.41} [/latex]

       
       

    2. [latex] 7^{\frac{2}{3}} = \sqrt[3]{7^2} = 3.66 [/latex]

       
       

35. 1. [latex] 10^{-\frac{1}{5}} = \displaystyle{\sqrt[5]{\frac{1}{10}} = 0.63} [/latex]

       
       

    2. [latex] 2^{-\frac{9}{7}} = \displaystyle{\sqrt[7]{\left(\frac{1}{2}\right)^9} = 0.41} [/latex]

       
       

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

       
       

    2. [latex] 7^{\frac{37}{24}} = \sqrt[24]{7^{37}} = 20.08 [/latex]

       
       

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

       
       

    2. [latex] 6^3 = 216 [/latex]

       
       

41. 1. [latex] 8^4 = 4,096 [/latex]

       
       

    2. [latex] 7^{-3} = 0.003 [/latex]

       
       

43. 1. [latex] 0.67 [/latex]

       
       

    2. [latex] 0.83 [/latex]

       
       

45. 1. [latex] 0.33 [/latex]

       
       

    2. [latex] 8 [/latex]

       
       

47. 1. [latex] 0.91 [/latex]

       
       

    2. [latex] 2.25 [/latex]

       
       

Exercise 3.3

1. 1. –5 < 0

      
      

   2. –2 < 6

      
      

3. 1. 8 > –3

      
      

   2. 1 > –2

      
      

5. 1. –6 > –8

      
      

   2. –5 < –2

      
      

7. 1. –8, –6, –5, 2, 5, 8

      
      

   2. –9, –8, –6, 3, 4, 7

      
      

9. 1. –8, –5, 3, 7, 9, 10

      
      

   2. –13, –8, –3, 2, 12, 15

      
      

11. 1. 16

       
       

    2. –3

       
       

13. 1. –5

       
       

    2. 9

       
       

15. 1. –3

       
       

    2. –5

       
       

17. 1. 50

       
       

    2. 5

       
       

19. 1. –18

       
       

    2. –4

       
       

21. 1. –20

       
       

    2. 6

       
       

23. 1. 12

       
       

    2. 14

       
       

25. 1. 7

       
       

    2. –3

       
       

27. 1. 30

       
       

    2. –24

       
       

29. 1. 8

       
       

    2. –9

       
       

31. 1. 2

       
       

    2. 17

       
       

33. 1. 70

       
       

    2. –72

       
       

35. 1. 265

       
       

    2. 30

       
       

37. –132

39. –111

41. 8

43. 9

45. 1. [latex] (-6)^8 = 1,679,616 [/latex]

       
       

    2. [latex] 8^3 = 512 [/latex]

       
       

47. 1. [latex] (-4)^2 = 16 [/latex]

       
       

    2. [latex] -5^2 = -25 [/latex]

       
       

49. 1. [latex] (-2)^1 = -2 [/latex]

       
       

    2. [latex] -(3^{-4}) = -0.01 [/latex]

       
       

51. 1. –23

       
       

    2. 17

       
       

53. 1. –18

       
       

    2. –15

       
       

Review Exercises 3

1. 7

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

5. 1. 2^2 = 4

      
      

   2. 5^3 = 125

      
      

7. 1. 3^3 = 27

      
      

   2. 6^1 = 6

      
      

9. 1. 27

      
      

   2. 0.2

      
      

11. 1. 400

       
       

    2. –10,000,000

       
       

13. 1. [latex] \displaystyle{\frac{1}{5} = 0.2} [/latex]

       
       

    2. [latex] \displaystyle{\frac{1}{7} = 0.\overline{142857}} [/latex]

       
       

    3. [latex] \displaystyle{\frac{8}{9} = 0.\overline{8}} [/latex]

       
       

15. 1. 49

       
       

    2. [latex] \displaystyle{\frac{5}{6} = 0.8\overline{3}} [/latex]

       
       

    3. [latex] \displaystyle{\frac{6}{5} = 1.2} [/latex]

       
       

17. 1. 3.98

       
       

    2. 50

       
       

19. 1. 96.57

       
       

    2. 24

       
       

21. 1. –19

       
       

    2. 27

       
       

23. 1. 7,180.08

       
       

    2. 1,817.28

       
       

25. 164,593.54

27. –17,668.97

29. 1. 0

       
       

    2. –7

       
       

31. 1. 13

       
       

    2. –28

       
       

33. 1. [latex] 5 [/latex]; [latex] 7.1011 \times 10^3 [/latex]

       
       

    2. [latex] 5 [/latex]; [latex] 5.4001 \times 10^1 [/latex]

       
       

    3. [latex] 2 [/latex]; [latex] 7.2 \times 10^{-3} [/latex]

       
       

35. 1. 890

       
       

    2. 0.056

       
       

    3. 0.000964

       
       

37. 1. [latex] 4.7495 \times 10^{14} [/latex]

       
       

    2. [latex] 7.715 \times 10^{-2} [/latex]

       
       

39. 1. [latex] 3.9744 \times 10^6 [/latex]

       
       

    2. [latex] 2.759 \times 10^{-3} [/latex]

       
       

41. 1. [latex] 2.4 \times 10^{12} [/latex]

       
       

    2. [latex] 4.5 \times 10^{-10} [/latex]

       
       

43. 1. [latex] 5.0 \times 10^{-4} [/latex]

       
       

    2. [latex] 2.5 \times 10^{-7} [/latex]

       
       

Self-Test Exercises 3

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

      
      

   2. [latex] 2^7 [/latex]

      
      

   3. [latex] 3^3 [/latex]

      
      

   4. [latex] 2^3 [/latex]

      
      

2. 1. [latex] 3^{10} [/latex]

      
      

   2. [latex] 10^9 [/latex]

      
      

3. 1. [latex] \displaystyle{\frac{3^2}{2^6} = 0.14} [/latex]

      
      

   2. [latex] 2 \times 3^2 = 18 [/latex]

      
      

4. 1. 0.01

      
      

   2. 1

      
      

5. 1. 0.84

      
      

   2. 5.80

      
      

6. 1. 13.6

      
      

   2. 1

      
      

7. 1. 32

      
      

   2. 5.80

      
      

8. 1. 1,953,125

      
      

   2. 3

      
      

9. 1. –26

      
      

   2. 42

      
      

10. 1. –61

       
       

    2. –16

       
       

11. 1. –3

       
       

    2. 0

       
       

12. 1. 2.67

       
       

    2. 0.19

       
       

13. 1. [latex] 1.009 \times 10^1 [/latex]

       
       

    2. [latex] 5.0 \times 10^{-3} [/latex]

       
       

    3. [latex] 6.02 \times 10^4 [/latex]

       
       

14. 1. 2,700

       
       

    2. 0.00415

       
       

    3. 0.030405

       
       

15. 1. [latex] 5.009 \times 10^4 [/latex]

       
       

    2. [latex] -9.534 \times 10^{-7} [/latex]

       
       

16. 1. [latex] 5.2 \times 10^{-1} [/latex]

       
       

    2. [latex] 1.4 \times 10^{19} [/latex]