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This is “Review Exercises and Sample Exam”, section 8.7 from the book Beginning Algebra (v. 1.0). For details on it (including licensing), click here.
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8.7 Review Exercises and Sample Exam
Review Exercises
(Assume all variables represent nonnegative numbers.)
Radicals
Simplify.
1. 36
2. 425
3. −16
4. −9
5. 1253
6. 3 −83
7. 1643
8. −5 −273
9. 40
10. −350
11. 9881
12. 1121
13. 5 1923
14. 2 −543
Simplifying Radical Expressions
Simplify.
15. 49x2
16. 25a2b2
17. 75x3y2
18. 200m4n3
19. 18x325y2
20. 108x349y4
21. 216x33
22. −125x6y33
23. 27a7b5c33
24. 120x9y43
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. 83+33
30. 1210−210
31. 143+52−53−62
32. 22ab−5ab+7ab−2ab
33. 7x−(3x+2y)
34. (8yx−7xy)−(5xy−12yx)
35. 45+12−20−75
36. 24−32+54−232
37. 23x2+45x−x27+20x
38. 56a2b+8a2b2−224a2b−a18b2
39. 5y4x2y−(x16y3−29x2y3)
40. (2b9a2c−3a16b2c)−(64a2b2c−9ba2c)
41. 216x3−125xy3−8x3
42. 128x33−2x⋅543+3 2x33
43. 8x3y3−2x⋅8y3+27x3y3+x⋅y3
44. 27a3b3−3 8ab33+a⋅64b3−b⋅a3
Multiplying and Dividing Radical Expressions
Multiply.
45. 3⋅6
46. (35)2
47. 2(3−6)
48. (2−6)2
49. (1−5)(1+5)
50. (23+5)(32−25)
51. 2a23⋅4a3
52. 25a2b3⋅5a2b23
Divide.
53. 724
54. 104864
55. 98x4y236x2
56. 81x6y738y33
Rationalize the denominator.
57. 27
58. 63
59. 142x
60. 1215
61. 12x23
62. 5a2b5ab23
63. 13−2
64. 2−62+6
Rational Exponents
Express in radical form.
65. 71/2
66. 32/3
67. x4/5
68. y−3/4
Write as a radical and then simplify.
69. 41/2
70. 501/2
71. 42/3
72. 811/3
73. (14)3/2
74. (1216)−1/3
Perform the operations and simplify. Leave answers in exponential form.
75. 31/2⋅33/2
76. 21/2⋅21/3
77. 43/241/2
78. 93/491/4
79. (36x4y2)1/2
80. (8x6y9)1/3
81. ( a 4/3 a 1/2)2/5
82. (16 x 4/3 y 2)1/2
Solving Radical Equations
Solve.
83. x=5
84. 2x−1=3
85. x−8+2=5
86. 3x−5−1=11
87. 5x−3=2x+15
88. 8x−15=x
89. x+41=x−1
90. 7−3x=x−3
91. 2(x+1)=2(x+1)
92. x(x+6)=4
93. x(3x+10)3=2
94. 2x2−x3+4=5
95. 3(x+4)(x+1)3=5x+373
96. 3x2−9x+243=(x+2)23
97. y1/2−3=0
98. y1/3+3=0
99. (x−5)1/2−2=0
100. (2x−1)1/3−5=0
Sample Exam
In problems 1–18, assume all variables represent nonnegative numbers.
1. Simplify.
1.100
2.−100
3.−100
2. Simplify.
1.273
2.−273
3.−273
3. 12825
4. 1921253
5. 512x2y3z
6. 250x2y3z53
Perform the operations.
7. 524−108+96−327
8. 38x2y−(x200y−18x2y)
9. 2ab(32a−b)
10. (x−2y)2
Rationalize the denominator.
11. 102x
12. 14xy23
13. 1x+5
14. 2−32+3
Perform the operations and simplify. Leave answers in exponential form.
15. 22/3⋅21/6
16. 104/5101/3
17. (121a4b2)1/2
18. (9 y 1/3 x 6)1/2y1/6
Solve.
19. x−7=0
20. 3x+5=1
21. 2x−1+2=x
22. 31−10x=x−4
23. (2x+1)(3x+2)=3(2x+1)
24. x(2x−15)3=3
25. The period, T, of a pendulum in seconds is given the formula T=2πL32, 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: 210
11: 729
13: 20 33
15: 7x
17: 5xy3x
19: 3x2x5y
21: 6x
23: 3a2bc⋅ab23
25: 13
27: 229
29: 113
31: 93−2
33: 4x−2y
35: 5−33
37: −x3+55x
39: 12xyy
41: 4 x3−5 xy3
43: 2x⋅y3
45: 32
47: 6−23
49: −4
51: 2a
53: 32
55: 7xy26
57: 277
59: 72xx
61: 4x32x
63: 3+2
65: 7
67: x45
69: 2
71: 2 23
73: 1/8
75: 9
77: 4
79: 6x2y
81: a1/3
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
1:
1.10
2.Not a real number
3.−10
3: 825
5: 10xy3yz
7: 146−153
9: 6a2b−2ba
11: 52xx
13: x−5x−25
15: 25/6
17: 11a2b
19: 49
21: 5
23: −1/2, 1/3
25: 1.8 feet
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