Review Exercises and Sample Exam

8.7 Review Exercises and Sample Exam

Review Exercises

(Assume all variables represent nonnegative numbers.)

Radicals

Simplify.

1. $\sqrt{36}$

2. $\sqrt{\frac{4}{25}}$

3. $\sqrt{- 16}$

4. $- \sqrt{9}$

5. $\sqrt[3]{125}$

6. $3 \sqrt[3]{- 8}$

7. $\sqrt[3]{\frac{1}{64}}$

8. $- 5 \sqrt[3]{- 27}$

9. $\sqrt{40}$

10. $- 3 \sqrt{50}$

11. $\sqrt{\frac{98}{81}}$

12. $\sqrt{\frac{1}{121}}$

13. $5 \sqrt[3]{192}$

14. $2 \sqrt[3]{- 54}$

Simplifying Radical Expressions

Simplify.

15. $\sqrt{49 x^{2}}$

16. $\sqrt{25 a^{2} b^{2}}$

17. $\sqrt{75 x^{3} y^{2}}$

18. $\sqrt{200 m^{4} n^{3}}$

19. $\sqrt{\frac{18 x^{3}}{25 y^{2}}}$

20. $\sqrt{\frac{108 x^{3}}{49 y^{4}}}$

21. $\sqrt[3]{216 x^{3}}$

22. $\sqrt[3]{- 125 x^{6} y^{3}}$

23. $\sqrt[3]{27 a^{7} b^{5} c^{3}}$

24. $\sqrt[3]{120 x^{9} y^{4}}$

Use the distance formula to calculate the distance between the given two points.

25. (5, −8) and (2, −10)

26. (−7, −1) and (−6, 1)

27. (−10, −1) and (0, −5)

28. (5, −1) and (−2, −2)

Adding and Subtracting Radical Expressions

Simplify.

29. $8 \sqrt{3} + 3 \sqrt{3}$

30. $12 \sqrt{10} - 2 \sqrt{10}$

31. $14 \sqrt{3} + 5 \sqrt{2} - 5 \sqrt{3} - 6 \sqrt{2}$

32. $22 \sqrt{a b} - 5 a \sqrt{b} + 7 \sqrt{a b} - 2 a \sqrt{b}$

33. $7 \sqrt{x} - (3 \sqrt{x} + 2 \sqrt{y})$

34. $(8 y \sqrt{x} - 7 x \sqrt{y}) - (5 x \sqrt{y} - 12 y \sqrt{x})$

35. $\sqrt{45} + \sqrt{12} - \sqrt{20} - \sqrt{75}$

36. $\sqrt{24} - \sqrt{32} + \sqrt{54} - 2 \sqrt{32}$

37. $2 \sqrt{3 x^{2}} + \sqrt{45 x} - x \sqrt{27} + \sqrt{20 x}$

38. $5 \sqrt{6 a^{2} b} + \sqrt{8 a^{2} b^{2}} - 2 \sqrt{24 a^{2} b} - a \sqrt{18 b^{2}}$

39. $5 y \sqrt{4 x^{2} y} - (x \sqrt{16 y^{3}} - 2 \sqrt{9 x^{2} y^{3}})$

40. $(2 b \sqrt{9 a^{2} c} - 3 a \sqrt{16 b^{2} c}) - (\sqrt{64 a^{2} b^{2} c} - 9 b \sqrt{a^{2} c})$

41. $\sqrt[3]{216 x} - \sqrt[3]{125 x y} - \sqrt[3]{8 x}$

42. $\sqrt[3]{128 x^{3}} - 2 x \cdot \sqrt[3]{54} + 3 \sqrt[3]{2 x^{3}}$

43. $\sqrt[3]{8 x^{3} y} - 2 x \cdot \sqrt[3]{8 y} + \sqrt[3]{27 x^{3} y} + x \cdot \sqrt[3]{y}$

44. $\sqrt[3]{27 a^{3} b} - 3 \sqrt[3]{8 a b^{3}} + a \cdot \sqrt[3]{64 b} - b \cdot \sqrt[3]{a}$

Multiplying and Dividing Radical Expressions

Multiply.

45. $\sqrt{3} \cdot \sqrt{6}$

46. ${(3 \sqrt{5})}^{2}$

47. $\sqrt{2} (\sqrt{3} - \sqrt{6})$

48. ${(\sqrt{2} - \sqrt{6})}^{2}$

49. $(1 - \sqrt{5}) (1 + \sqrt{5})$

50. $(2 \sqrt{3} + \sqrt{5}) (3 \sqrt{2} - 2 \sqrt{5})$

51. $\sqrt[3]{2 a^{2}} \cdot \sqrt[3]{4 a}$

52. $\sqrt[3]{25 a^{2} b} \cdot \sqrt[3]{5 a^{2} b^{2}}$

Divide.

53. $\frac{\sqrt{72}}{\sqrt{4}}$

54. $\frac{10 \sqrt{48}}{\sqrt{64}}$

55. $\frac{\sqrt{98 x^{4} y^{2}}}{\sqrt{36 x^{2}}}$

56. $\frac{\sqrt[3]{81 x^{6} y^{7}}}{\sqrt[3]{8 y^{3}}}$

Rationalize the denominator.

57. $\frac{2}{\sqrt{7}}$

58. $\frac{\sqrt{6}}{\sqrt{3}}$

59. $\frac{14}{\sqrt{2 x}}$

60. $\frac{1}{2 \sqrt{15}}$

61. $\frac{1}{\sqrt[3]{2 x^{2}}}$

62. $\frac{5 a^{2} b}{\sqrt[3]{5 a b^{2}}}$

63. $\frac{1}{\sqrt{3} - \sqrt{2}}$

64. $\frac{\sqrt{2} - \sqrt{6}}{\sqrt{2} + \sqrt{6}}$

Rational Exponents

Express in radical form.

65. $7^{1 / 2}$

66. $3^{2 / 3}$

67. $x^{4 / 5}$

68. $y^{- 3 / 4}$

Write as a radical and then simplify.

69. $4^{1 / 2}$

70. $50^{1 / 2}$

71. $4^{2 / 3}$

72. $81^{1 / 3}$

73. ${(\frac{1}{4})}^{3 / 2}$

74. ${(\frac{1}{216})}^{- 1 / 3}$

Perform the operations and simplify. Leave answers in exponential form.

75. $3^{1 / 2} \cdot 3^{3 / 2}$

76. $2^{1 / 2} \cdot 2^{1 / 3}$

77. $\frac{4^{3 / 2}}{4^{1 / 2}}$

78. $\frac{9^{3 / 4}}{9^{1 / 4}}$

79. ${(36 x^{4} y^{2})}^{1 / 2}$

80. ${(8 x^{6} y^{9})}^{1 / 3}$

81. ${(\frac{a^{4 / 3}}{a^{1 / 2}})}^{2 / 5}$

82. ${(\frac{16 x^{4 / 3}}{y^{2}})}^{1 / 2}$

Solving Radical Equations

Solve.

83. $\sqrt{x} = 5$

84. $\sqrt{2 x - 1} = 3$

85. $\sqrt{x - 8} + 2 = 5$

86. $3 \sqrt{x - 5} - 1 = 11$

87. $\sqrt{5 x - 3} = \sqrt{2 x + 15}$

88. $\sqrt{8 x - 15} = x$

89. $\sqrt{x + 41} = x - 1$

90. $\sqrt{7 - 3 x} = x - 3$

91. $2 (x + 1) = \sqrt{2 (x + 1)}$

92. $\sqrt{x (x + 6)} = 4$

93. $\sqrt[3]{x (3 x + 10)} = 2$

94. $\sqrt[3]{2 x^{2} - x} + 4 = 5$

95. $\sqrt[3]{3 (x + 4) (x + 1)} = \sqrt[3]{5 x + 37}$

96. $\sqrt[3]{3 x^{2} - 9 x + 24} = \sqrt[3]{{(x + 2)}^{2}}$

97. $y^{1 / 2} - 3 = 0$

98. $y^{1 / 3} + 3 = 0$

99. ${(x - 5)}^{1 / 2} - 2 = 0$

100. ${(2 x - 1)}^{1 / 3} - 5 = 0$

Sample Exam

In problems 1–18, assume all variables represent nonnegative numbers.

1. Simplify.

$\sqrt{100}$
$\sqrt{- 100}$
$- \sqrt{100}$

2. Simplify.

$\sqrt[3]{27}$
$\sqrt[3]{- 27}$
$- \sqrt[3]{27}$

3. $\sqrt{\frac{128}{25}}$

4. $\sqrt[3]{\frac{192}{125}}$

5. $5 \sqrt{12 x^{2} y^{3} z}$

6. $\sqrt[3]{250 x^{2} y^{3} z^{5}}$

Perform the operations.

7. $5 \sqrt{24} - \sqrt{108} + \sqrt{96} - 3 \sqrt{27}$

8. $3 \sqrt{8 x^{2} y} - (x \sqrt{200 y} - \sqrt{18 x^{2} y})$

9. $2 \sqrt{a b} (3 \sqrt{2 a} - \sqrt{b})$

10. ${(\sqrt{x} - 2 \sqrt{y})}^{2}$

Rationalize the denominator.

11. $\frac{10}{\sqrt{2 x}}$

12. $\frac{1}{\sqrt[3]{4 x y^{2}}}$

13. $\frac{1}{\sqrt{x} + 5}$

14. $\frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}}$

Perform the operations and simplify. Leave answers in exponential form.

15. $2^{2 / 3} \cdot 2^{1 / 6}$

16. $\frac{10^{4 / 5}}{10^{1 / 3}}$

17. ${(121 a^{4} b^{2})}^{1 / 2}$

18. $\frac{{(9 y^{1 / 3} x^{6})}^{1 / 2}}{y^{1 / 6}}$

Solve.

19. $\sqrt{x} - 7 = 0$

20. $\sqrt{3 x + 5} = 1$

21. $\sqrt{2 x - 1} + 2 = x$

22. $\sqrt{31 - 10 x} = x - 4$

23. $\sqrt{(2 x + 1) (3 x + 2)} = \sqrt{3 (2 x + 1)}$

24. $\sqrt[3]{x (2 x - 15)} = 3$

25. The period, T, of a pendulum in seconds is given the formula $T = 2 π \sqrt{\frac{L}{32}}$ , where L represents the length in feet. Calculate the length of a pendulum if the period is 1½ seconds. Round off to the nearest tenth.

Review Exercises Answers

1: 6

3: Not a real number

5: $5$

7: 1/4

9: $2 \sqrt{10}$

11: $\frac{7 \sqrt{2}}{9}$

13: $20 \sqrt[3]{3}$

15: $7 x$

17: $5 x y \sqrt{3 x}$

19: $\frac{3 x \sqrt{2 x}}{5 y}$

21: $6 x$

23: $3 a^{2} b c \cdot \sqrt[3]{a b^{2}}$

25: $\sqrt{13}$

27: $2 \sqrt{29}$

29: $11 \sqrt{3}$

31: $9 \sqrt{3} - \sqrt{2}$

33: $4 \sqrt{x} - 2 \sqrt{y}$

35: $\sqrt{5} - 3 \sqrt{3}$

37: $- x \sqrt{3} + 5 \sqrt{5 x}$

39: $12 x y \sqrt{y}$

41: $4 \sqrt[3]{x} - 5 \sqrt[3]{x y}$

43: $2 x \cdot \sqrt[3]{y}$

45: $3 \sqrt{2}$

47: $\sqrt{6} - 2 \sqrt{3}$

49: −4

51: $2 a$

53: $3 \sqrt{2}$

55: $\frac{7 x y \sqrt{2}}{6}$

57: $\frac{2 \sqrt{7}}{7}$

59: $\frac{7 \sqrt{2 x}}{x}$

61: $\frac{\sqrt[3]{4 x}}{2 x}$

63: $\sqrt{3} + \sqrt{2}$

65: $\sqrt{7}$

67: $\sqrt[5]{x^{4}}$

69: 2

71: $2 \sqrt[3]{2}$

73: 1/8

75: 9

77: 4

79: $6 x^{2} y$

81: $a^{1 / 3}$

83: 25

85: 17

87: 6

89: 8

91: −1/2, −1

93: 2/3, −4

95: −5, 5/3

97: 9

99: 9

Sample Exam Answers

10
Not a real number
−10

3: $\frac{8 \sqrt{2}}{5}$

5: $10 x y \sqrt{3 y z}$

7: $14 \sqrt{6} - 15 \sqrt{3}$

9: $6 a \sqrt{2 b} - 2 b \sqrt{a}$

11: $\frac{5 \sqrt{2 x}}{x}$

13: $\frac{\sqrt{x} - 5}{x - 25}$

15: $2^{5 / 6}$

17: $11 a^{2} b$

19: 49

21: 5

23: −1/2, 1/3

25: 1.8 feet