Review Exercises

Roots and Radicals

Simplify.

1. $-\sqrt{121}$

2. $\sqrt{{\left(\text{−}7\right)}^2}$

3. $\sqrt{{\left(xy\right)}^2}$

4. $\sqrt{{\left(6x-7\right)}^2}$

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

6. $\sqrt[3]{\text{−}27}$

7. $\sqrt[3]{{\left(xy\right)}^3}$

8. $\sqrt[3]{{\left(6x+1\right)}^3}$

9. Given $f\left(x\right)=\sqrt{x+10}$, find $f\left(\text{−}1\right)$ and $f\left(6\right).$

10. Given $g\left(x\right)=\sqrt[3]{x-5}$, find $g\left(4\right)$ and $g\left(13\right).$

11. Determine the domain of the function defined by $g\left(x\right)=\sqrt{5x+2}.$

12. Determine the domain of the function defined by $g\left(x\right)=\sqrt[3]{3x-1}.$

Simplify.

13. $\sqrt[3]{250}$

14. $4\sqrt[3]{120}$

15. $\text{−}3\sqrt[3]{108}$

16. $$10\sqrt[5]{\frac{1}{32}}$$
17. $$\text{−}6\sqrt[4]{\frac{81}{16}}$$

18. $\sqrt[6]{128}$

19. $\sqrt[5]{\text{−}192}$

20. $\text{−}3\sqrt{420}$

Simplifying Radical Expressions

Simplify.

21. $\sqrt{20x^4{y}^3}$

22. $\text{−}4\sqrt{54x^6{y}^3}$

23. $\sqrt{x^2-14x+49}$

24. $\sqrt{{\left(x-8\right)}^4}$

Simplify. (Assume all variable expressions are nonzero.)

25. $\sqrt{100x^2{y}^4}$

26. $\sqrt{36a^6{b}^2}$

27. $$\sqrt{\frac{8a^2}{b^4}}$$
28. $$\sqrt{\frac{72x^{4}y}{z^6}}$$

29. $10x\sqrt{150x^7{y}^4}$

30. $\text{−}5n^2\sqrt{25m^{10}{n}^6}$

31. $\sqrt[3]{48x^6{y}^3{z}^2}$

32. $\sqrt[3]{270a^{10}{b}^8{c}^3}$

33. $$\sqrt[3]{\frac{a^3{b}^5}{64c^6}}$$
34. $$\sqrt[5]{\frac{a^{26}}{32b^5{c}^{10}}}$$

35. The period T in seconds of a pendulum is given by the formula $T=2\pi \sqrt{\frac{L}{32}}$ where L represents the length in feet of the pendulum. Calculate the period of a pendulum that is $2\frac{1}{2}$ feet long. Give the exact answer and the approximate answer to the nearest hundredth of a second.

36. The time in seconds an object is in free fall is given by the formula $t=\frac{\sqrt{s}}{4}$ where s represents the distance in feet the object has fallen. How long does it take an object to fall 28 feet? Give the exact answer and the approximate answer to the nearest tenth of a second.

37. Find the distance between (−5, 6) and (−3,−4).

38. Find the distance between $\left(\frac{2}{3},-\frac{1}{2}\right)$ and $\left(1,-\frac{3}{4}\right).$

Determine whether or not the three points form a right triangle. Use the Pythagorean theorem to justify your answer.

39. (−4,5), (−3,−1), and (3,0)

40. (−1,−1), (1,3), and (−6,1)

Adding and Subtracting Radical Expressions

Simplify. Assume all radicands containing variables are nonnegative.

41. $7\sqrt{2}+5\sqrt{2}$

42. $8\sqrt{15}-2\sqrt{15}$

43. $14\sqrt{3}+5\sqrt{2}-5\sqrt{3}-6\sqrt{2}$

44. $22\sqrt{ab}-5a\sqrt{b}+7\sqrt{ab}-2a\sqrt{b}$

45. $7\sqrt{x}-\left(3\sqrt{x}+2\sqrt{y}\right)$

46. $\left(8y\sqrt{x}-7x\sqrt{y}\right)-\left(5x\sqrt{y}-12y\sqrt{x}\right)$

47. $\left(3\sqrt{5}+2\sqrt{6}\right)+\left(8\sqrt{5}-3\sqrt{6}\right)$

48. $\left(4\sqrt[3]{3}-\sqrt[3]{12}\right)-\left(5\sqrt[3]{3}-2\sqrt[3]{12}\right)$

49. $\left(2-\sqrt{10x}+3\sqrt{y}\right)-\left(1+2\sqrt{10x}-6\sqrt{y}\right)$

50. $$\left(3a\sqrt[3]{a{b}^2}+6\sqrt[3]{a^{2}b}\right)+\left(9a\sqrt[3]{a{b}^2}-12\sqrt[3]{a^{2}b}\right)$$

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

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

53. $2\sqrt{3x^2}+\sqrt{45x}-x\sqrt{27}+\sqrt{20x}$

54. $5\sqrt{6a^{2}b}+\sqrt{8a^2{b}^2}-2\sqrt{24a^{2}b}-a\sqrt{18b^2}$

55. $5y\sqrt{4x^{2}y}-\left(x\sqrt{16y^3}-2\sqrt{9x^2{y}^3}\right)$

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

57. $\sqrt[3]{216x}-\sqrt[3]{125xy}-\sqrt[3]{8x}$

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

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

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

61. Calculate the perimeter of the triangle formed by the following set of vertices: $\left\{\left(\text{−}3,\text{−}2\right),\left(\text{−}1,1\right),\left(1,\text{−}2\right)\right\}.$

62. Calculate the perimeter of the triangle formed by the following set of vertices: $\left\{\left(0,\text{−}4\right),\left(2,0\right),\left(\text{−}3,0\right)\right\}.$

Multiplying and Dividing Radical Expressions

Multiply.

63. $\sqrt{6}\cdot \sqrt{15}$

64. ${\left(4\sqrt{2}\right)}^2$

65. $\sqrt{2}\left(\sqrt{2}-\sqrt{10}\right)$

66. ${\left(\sqrt{5}-\sqrt{6}\right)}^2$

67. $\left(5-\sqrt{3}\right)\left(5+\sqrt{3}\right)$

68. $\left(2\sqrt{6}+\sqrt{3}\right)\left(\sqrt{2}-5\sqrt{3}\right)$

69. ${\left(\sqrt{a}-5\sqrt{b}\right)}^2$

70. $3\sqrt{xy}\left(\sqrt{x}-2\sqrt{y}\right)$

71. $\sqrt[3]{3a^2}\cdot \sqrt[3]{18a}$

72. $\sqrt[3]{49a^{2}b}\cdot \sqrt[3]{7a^2{b}^2}$

Divide. Assume all variables represent nonzero numbers and rationalize the denominator where appropriate.

73. $$\frac{\sqrt{72}}{\sqrt{9}}$$
74. $$\frac{10\sqrt{48}}{\sqrt{64}}$$
75. $$\frac{5}{\sqrt{5}}$$
76. $$\frac{\sqrt{15}}{\sqrt{2}}$$
77. $$\frac{3}{2\sqrt{6}}$$
78. $$\frac{2+\sqrt{5}}{\sqrt{10}}$$
79. $$\frac{18}{\sqrt{3x}}$$
80. $$\frac{2\sqrt{3x}}{\sqrt{6xy}}$$
81. $$\frac{1}{\sqrt[3]{3x^2}}$$
82. $$\frac{5a{b}^2}{\sqrt[3]{5a^{2}b}}$$
83. $$\sqrt[3]{\frac{5x{z}^2}{49x^2{y}^{2}z}}$$
84. $$\frac{1}{\sqrt[5]{8x^4{y}^{2}z}}$$
85. $$\frac{9x^{2}y}{\sqrt[5]{81x{y}^2{z}^3}}$$
86. $$\sqrt[5]{\frac{27a{b}^3}{15a^{4}b{c}^2}}$$
87. $$\frac{1}{\sqrt{5}-\sqrt{3}}$$
88. $$\frac{\sqrt{3}}{\sqrt{2}+1}$$
89. $$\frac{\text{−}3\sqrt{6}}{2-\sqrt{10}}$$
90. $$\frac{\sqrt{xy}}{\sqrt{x}-\sqrt{y}}$$
91. $$\frac{\sqrt{2}-\sqrt{6}}{\sqrt{2}+\sqrt{6}}$$
92. $$\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}$$

93. The base of a triangle measures $2\sqrt{6}$ units and the height measures $3\sqrt{15}$ units. Find the area of the triangle.

94. If each side of a square measures $5+2\sqrt{10}$ units, find the area of the square.

Rational Exponents

Express in radical form.

95. ${11}^{1/2}$

96. $2^{2/3}$

97. $x^{3/5}$

98. $a^{\text{−}4/5}$

Write as a radical and then simplify.

99. ${16}^{1/2}$

100. ${72}^{1/2}$

101. $8^{2/3}$

102. ${32}^{1/3}$

103. $${\left(\frac{1}{9}\right)}^{3/2}$$
104. $${\left(\frac{1}{216}\right)}^{\text{−}1/3}$$

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

105. $6^{1/2}\cdot 6^{3/2}$

106. $3^{1/3}\cdot 3^{1/2}$

107. $$\frac{6^{5/2}}{6^{3/2}}$$
108. $$\frac{4^{3/4}}{4^{1/4}}$$

109. ${\left(64x^6{y}^2\right)}^{1/2}$

110. ${\left(27x^{12}{y}^6\right)}^{1/3}$

111. $${\left(\frac{a^{4/3}}{a^{1/2}}\right)}^{2/5}$$
112. $${\left(\frac{16x^{4/3}}{y^2}\right)}^{1/2}$$
113. $$\frac{56x^{3/4}{y}^{3/2}}{14x^{1/2}{y}^{2/3}}$$
114. $$\frac{{\left(4a^4{b}^{2/3}{c}^{4/3}\right)}^{1/2}}{2a^2{b}^{1/6}{c}^{2/3}}$$

115. ${\left(9x^{\text{−}4/3}{y}^{1/3}\right)}^{\text{−}3/2}$

116. ${\left(16x^{\text{−}4/5}{y}^{1/2}{z}^{\text{−}2/3}\right)}^{\text{−}3/4}$

Perform the operations with mixed indices.

117. $\sqrt{y}\cdot \sqrt[5]{y^2}$

118. $\sqrt[3]{y}\cdot \sqrt[5]{y^3}$

119. $\frac{\sqrt[3]{y^2}}{\sqrt[5]{y}}$

120. $\sqrt{\sqrt[3]{y^2}}$

Solving Radical Equations

Solve.

121. $2\sqrt{x}+3=13$

122. $\sqrt{3x-2}=4$

123. $\sqrt{x-5}+4=8$

124. $5\sqrt{x+3}+7=2$

125. $\sqrt{4x-3}=\sqrt{2x+15}$

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

127. $x-1=\sqrt{13-x}$

128. $\sqrt{4x-3}=2x-3$

129. $\sqrt{x+5}=5-\sqrt{x}$

130. $\sqrt{x+3}=3\sqrt{x}-1$

131. $\sqrt{2\left(x+1\right)}-\sqrt{x+2}=1$

132. $\sqrt{6-x}+\sqrt{x-2}=2$

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

134. $\sqrt{9-x}=\sqrt{x+16}-1$

135. $\sqrt[3]{4x-3}=2$

136. $\sqrt[3]{x-8}=\text{−}1$

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

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

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

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

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

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

143. ${\left(x-5\right)}^{1/2}-2=0$

144. ${\left(2x-1\right)}^{1/3}-5=0$

145. ${\left(x-1\right)}^{1/2}=x^{1/2}-1$

146. ${\left(x-2\right)}^{1/2}-{\left(x-6\right)}^{1/2}=2$

147. ${\left(x+4\right)}^{1/2}-{\left(3x\right)}^{1/2}=\text{−}2$

148. ${\left(5x+6\right)}^{1/2}=3-{\left(x+3\right)}^{1/2}$

149. Solve for g: $t=\sqrt{\frac{2s}{g}}.$

150. Solve for x: $y=\sqrt[3]{x+4}-2.$

151. The period in seconds of a pendulum is given by the formula $T=2\pi \sqrt{\frac{L}{32}}$ where L represents the length in feet of the pendulum. Find the length of a pendulum that has a period of $1\frac{1}{2}$ seconds. Find the exact answer and the approximate answer rounded off to the nearest tenth of a foot.

152. The outer radius of a spherical shell is given by the formula $r=\sqrt[3]{\frac{3V}{4\pi }}+2$ where V represents the inner volume in cubic centimeters. If the outer radius measures 8 centimeters, find the inner volume of the sphere.

153. The speed of a vehicle before the brakes are applied can be estimated by the length of the skid marks left on the road. On dry pavement, the speed v in miles per hour can be estimated by the formula $v=2\sqrt{6d}$, where d represents the length of the skid marks in feet. Estimate the length of a skid mark if the vehicle is traveling 30 miles per hour before the brakes are applied.

154. Find the real root of the function defined by $f\left(x\right)=\sqrt[3]{x-3}+2.$

Complex Numbers and Their Operations

Write the complex number in standard form $a+bi.$

155. $5-\sqrt{\text{−}16}$

156. $-\sqrt{\text{−}25}-6$

157. $$\frac{3+\sqrt{\text{−}8}}{10}$$
158. $$\frac{\sqrt{\text{−}12}-4}{6}$$

Perform the operations.

159. $\left(6-12i\right)+\left(4+7i\right)$

160. $\left(\text{−}3+2i\right)-\left(6-4i\right)$

161. $$\left(\frac{1}{2}-i\right)-\left(\frac{3}{4}-\frac{3}{2}i\right)$$
162. $$\left(\frac{5}{8}-\frac{1}{5}i\right)+\left(\frac{3}{2}-\frac{2}{3}i\right)$$

163. $\left(5-2i\right)-\left(6-7i\right)+\left(4-4i\right)$

164. $\left(10-3i\right)+\left(20+5i\right)-\left(30-15i\right)$

165. $4i\left(2-3i\right)$

166. $\left(2+3i\right)\left(5-2i\right)$

167. ${\left(4+i\right)}^2$

168. ${\left(8-3i\right)}^2$

169. $\left(3+2i\right)\left(3-2i\right)$

170. $\left(\text{−}1+5i\right)\left(\text{−}1-5i\right)$

171. $$\frac{2+9i}{2i}$$
172. $$\frac{i}{1-2i}$$
173. $$\frac{4+5i}{2-i}$$
174. $$\frac{3-2i}{3+2i}$$

175. $10-5{\left(2-3i\right)}^2$

176. ${\left(2-3i\right)}^2-\left(2-3i\right)+4$

177. $${\left(\frac{1}{1-i}\right)}^2$$
178. $${\left(\frac{1+2i}{3i}\right)}^2$$

179. $\sqrt{\text{−}8}\left(\sqrt{3}-\sqrt{\text{−}4}\right)$

180. $\left(1-\sqrt{\text{−}18}\right)\left(3-\sqrt{\text{−}2}\right)$

181. ${\left(\sqrt{\text{−}5}-\sqrt{\text{−}10}\right)}^2$

182. ${\left(1-\sqrt{\text{−}2}\right)}^2-{\left(1+\sqrt{\text{−}2}\right)}^2$

183. Show that both $\text{−}5i$ and $5i$ satisfy $x^2+25=0.$

184. Show that both $1-2i$ and $1+2i$ satisfy $x^2-2x+5=0.$

Answers

1. −11

3. $\left|xy\right|$

5. 5

7. $xy$

9. $f\left(\text{−}1\right)=3$; $f\left(6\right)=4$

11. $$\left[-\frac{2}{5},\infty \right)$$

13. $5\sqrt[3]{2}$

15. $\text{−}9\sqrt[3]{4}$

17. −9

19. $\text{−}2\sqrt[5]{6}$

21. $2x^2\left|y\right|\sqrt{5y}$

23. $\left|x-7\right|$

25. $10x{y}^2$

27. $$\frac{2a\sqrt{2}}{b^2}$$

29. $50x^4{y}^2\sqrt{6x}$

31. $2x^{2}y\sqrt[3]{6z^2}$

33. $$\frac{ab\sqrt[3]{b^2}}{4c^2}$$

35. $\frac{\pi \sqrt{5}}{4}$ seconds; 1.76 seconds

37. $2\sqrt{26}$ units

39. Right triangle

41. $12\sqrt{2}$

43. $9\sqrt{3}-\sqrt{2}$

45. $4\sqrt{x}-2\sqrt{y}$

47. $11\sqrt{5}-\sqrt{6}$

49. $1-3\sqrt{10x}+9\sqrt{y}$

51. $\sqrt{5}-3\sqrt{3}$

53. $-x\sqrt{3}+5\sqrt{5x}$

55. $12xy\sqrt{y}$

57. $4\sqrt[3]{x}-5\sqrt[3]{xy}$

59. $2x\sqrt[3]{y}$

61. $4+2\sqrt{13}$ units

63. $3\sqrt{10}$

65. $2-2\sqrt{5}$

67. 22

69. $a-10\sqrt{ab}+25b$

71. $3a\sqrt[3]{2}$

73. $2\sqrt{2}$

75. $\sqrt{5}$

77. $$\frac{\sqrt{6}}{4}$$
79. $$\frac{6\sqrt{3x}}{x}$$
81. $$\frac{\sqrt[3]{9x}}{3x}$$
83. $$\frac{\sqrt[3]{35x^{2}yz}}{7xy}$$
85. $$\frac{3xy\sqrt[5]{3x^4{y}^3{z}^2}}{z}$$
87. $$\frac{\sqrt{5}+\sqrt{3}}{2}$$

89. $\sqrt{6}+\sqrt{15}$

91. $\text{−}2+\sqrt{3}$

93. $9\sqrt{10}$ square units

95. $\sqrt{11}$

97. $\sqrt[5]{x^3}$

99. 4

101. $4$

103. 1/27

105. 36

107. 6

109. $8x^{3}y$

111. $a^{1/3}$

113. $4x^{1/4}{y}^{5/6}$

115. $$\frac{x^2}{27y^{1/2}}$$

117. $\sqrt[10]{y^9}$

119. $\sqrt[15]{y^7}$

121. 25

123. 21

125. 9

127. 4

129. 4

131. 7

133. 1

135. $\frac{11}{4}$

137. $\text{−}4,\frac{2}{3}$

139. $\text{−}5,\frac{5}{3}$

141. 9

143. 9

145. 1

147. 12

149. $g=\frac{2s}{t^2}$

151. $\frac{18}{{\pi }^2}$ feet; 1.8 feet

153. 37.5 feet

155. $5-4i$

157. $$\frac{3}{10}+\frac{\sqrt{2}}{5}i$$

159. $10-5i$

161. $-\frac{1}{4}+\frac{1}{2}i$

163. $3+i$

165. $12+8i$

167. $15+8i$

169. 13

171. $\frac{9}{2}-i$

173. $\frac{3}{5}+\frac{14}{5}i$

175. $35+60i$

177. $\frac{1}{2}i$

179. $4\sqrt{2}+2i\sqrt{6}$

181. $\text{−}15+10\sqrt{2}$

183. Answer may vary

Sample Exam

Simplify. (Assume all variables are positive.)

1. $5x\sqrt{121x^2{y}^4}$

2. $2x{y}^2\sqrt[3]{\text{−}64x^6{y}^9}$

3. Calculate the distance between $\left(\text{−}5,\text{−}3\right)$ and $\left(\text{−}2,6\right).$

4. The time in seconds an object is in free fall is given by the formula $t=\frac{\sqrt{s}}{4}$ where s represents the distance in feet that the object has fallen. If a stone is dropped into a 36-foot pit, how long will it take to hit the bottom of the pit?

Perform the operations and simplify. (Assume all variables are positive and rationalize the denominator where appropriate.)

5. $\sqrt{150x{y}^2}-2\sqrt{18x^3}+y\sqrt{24x}+x\sqrt{128x}$

6. $$3\sqrt[3]{16x^3{y}^2}-\left(2x\sqrt[3]{250y^2}-\sqrt[3]{54x^3{y}^2}\right)$$

7. $2\sqrt{2}\left(\sqrt{2}-3\sqrt{6}\right)$

8. ${\left(\sqrt{10}-\sqrt{5}\right)}^2$

9. $$\frac{\sqrt{6}}{\sqrt{2}+\sqrt{3}}$$
10. $$\frac{2x}{\sqrt{2xy}}$$
11. $$\frac{1}{\sqrt[5]{8x{y}^2{z}^4}}$$

12. Simplify: ${81}^{3/4}.$

13. Express in radical form: $x^{\text{−}3/5}.$

Simplify. Assume all variables are nonzero and leave answers in exponential form.

14. ${\left(81x^4{y}^2\right)}^{\text{−}1/2}$

15. $$\frac{{\left(25a^{4/3}{b}^8\right)}^{3/2}}{a^{1/2}b}$$

Solve.

16. $\sqrt{x}-5=1$

17. $\sqrt[3]{5x-2}+6=4$

18. $5\sqrt{2x+5}-2x=11$

19. $\sqrt{4-3x}+2=x$

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

21. The time in seconds an object is in free fall is given by the formula $t=\frac{\sqrt{s}}{4}$ where s represents the distance in feet that the object has fallen. If a stone is dropped into a pit and it takes 4 seconds to reach the bottom, how deep is the pit?

22. The width in inches of a container is given by the formula $w=\frac{\sqrt[3]{4V}}{2}+1$ where V represents the inside volume in cubic inches of the container. What is the inside volume of the container if the width is 6 inches?

Perform the operations and write the answer in standard form.

23. $\sqrt{\text{−}3}\left(\sqrt{6}-\sqrt{\text{−}3}\right)$

24. $\frac{4+3i}{2-i}$

25. $6-3{\left(2-3i\right)}^2$

Answers

1. $55x^2{y}^2$

3. $3\sqrt{10}$ units

5. $7y\sqrt{6x}+2x\sqrt{2x}$

7. $4-12\sqrt{3}$

9. $\text{−}2\sqrt{3}+3\sqrt{2}$

11. $$\frac{\sqrt[5]{4x^4{y}^{3}z}}{2xyz}$$
13. $$\frac{1}{\sqrt[5]{x^3}}$$

15. $125a^{3/2}{b}^{11}$

17. $-\frac{6}{5}$

19. Ø

21. 256 feet

23. $3+3i\sqrt{2}$

25. $21+36i$