0.3e: Exercises - Square Roots
- Page ID
- 38222
A: Simplify Square Roots
Exercise \(\PageIndex{1}\)
\( \bigstar \) Simplify. (Assume all variable expressions represent positive numbers, so absolute values is not needed.)
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- Answers to Odd Exercises:
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1. \(3 x \\[6pt]\)
3. \(6 a ^ { 2 }\)5. \(2 a ^ { 3 } \\[6pt]\)
7. \(3 a ^ { 2 } b ^ { 2 } \sqrt { 2 b }\)9. \(7a\\[6pt]\)
11. \(xy\)13. \(6 x \sqrt { 5 x }\\[6pt]\)
15. \(7 a b \sqrt { a }\)17. \(3 x ^ { 2 } y \sqrt { 5 x y }\\[6pt]\)
19. \(8 r s ^ { 3 } t ^ { 2 } \sqrt { t }\)
\( \bigstar \) Simplify. Rationalize denominators. (Assume all variable expressions represent positive numbers, so absolute values is not needed.)
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- Answers to Odd Exercises:
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21. \( 5 x - 4 \)
23. \( x - 3 \)
25. \( 2x + 3 \)27. \(x + 1\)
29. \(2 ( 3 x - 1 )\)
31. \(-6x \)33. \(- 10 x ^ { 2 } \sqrt { y }\)
35. \(12 a ^ { 3 } b ^ { 2 } \sqrt { a b }\)37. \(\dfrac { 3 x \sqrt { x } } { 5 y }\) 39. \(\dfrac { m ^ { 3 } \sqrt { m } } { 6 n ^ { 2 } }\) 41. \(\dfrac { r s ^ { 2 } \sqrt { 2 s } } { 5 t ^ { 2 } }\)
B: Add and Subtract Square Roots of Constants.
Exercise \(\PageIndex{2}\)
\( \bigstar \) Combine like terms
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- Answers to Odd Exercises:
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45. \(5 \sqrt { 3 }\\[6pt]\)
47. \(14 \sqrt { 3 }\)49. \(- 5 \sqrt { 5 }\\[6pt]\)
51. \(- \sqrt { 6 }\)53. \(8 \sqrt { 7 } - \sqrt { 2 }\\[6pt]\)
55. \(9 \sqrt { 5 } - 4 \sqrt { 3 }\)57. \(- \sqrt { 5 } - 4 \sqrt { 10 }\\[6pt]\)
\( \bigstar \) Simplify and combine like terms
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- Answers to Odd Exercises:
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59. \(3 \sqrt { 3 }\\[6pt]\)
61. \(2 \sqrt { 2 } + 3 \sqrt { 3 }\)63. \(5 \sqrt { 7 } - 5 \sqrt { 3 }\\[6pt]\)
65. \(5 \sqrt { 5 }\)67. \(10 \sqrt { 2 } - 3 \sqrt { 3 }\\[6pt]\)
69. \(2 \sqrt { 3 }\)71. \(23 \sqrt { 3 } - 6 \sqrt { 2 }\\[6pt]\)
73. \(- 8 \sqrt { 2 } + \sqrt { 3 }\)75. \(8 \sqrt { 3 } - 6 \sqrt { 6 }\)
C: Add and Subtract Square Root Expressions.
Exercise \( \PageIndex{3} \)
\( \bigstar \) Add. (Assume all radicands containing variable expressions are positive.)
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- Answers to Odd Exercises:
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79. \(- 3 \sqrt { 2 x }\) 81. \(16 \sqrt { x }\) 83. \(5 x \sqrt { y }\) 85. \(8 \sqrt { a b } - 15 \sqrt { a }\) 87. \(9 \sqrt { x y }\) 89. \(2 \sqrt { 2 x } + 6 \sqrt { 3 x }\)
\( \bigstar \) Simplify and add. (Assume all radicands containing variable expressions are positive.)
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- Answers to Odd Exercises:
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91. \(11 \sqrt { b }\\[6pt]\)
93. \(- 3 a \sqrt { b }\\[6pt]\)
95. \(8 \sqrt { x } - 5 \sqrt { y }\)97. \(20 \sqrt { 2 x } - 12 \sqrt { y }\\[6pt]\)
99. \(- 8 m \sqrt { n }\\[6pt]\)
101. \(- 2 x \sqrt { y } - 2 y \sqrt { x }\)103. \(- 4 x \sqrt { y }\\[6pt]\)
105. \(3 m ^ { 2 } \sqrt { 3 n }\\[6pt]\)
107. \(2 a \sqrt { 3 a b } - 12 a ^ { 2 } \sqrt { a b }\)
D: Multiply and Divide Square Root Expressions.
Exercise \(\PageIndex{4}\)
\( \bigstar \) Multiply and simplify. (Assume all variables represent non-negative real numbers.)
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- Answers to Odd Exercises:
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111. \(\sqrt{21}\\[6pt]\)
113. \(6\sqrt{2}\)115. \(2\sqrt{3}\\[6pt]\)
117. \(7\)119. \(70\sqrt{2}\\[6pt]\)
121. \(20\)123. \(2x\\[6pt]\)
125. \(6\sqrt{a}\)127. \(24x\sqrt{3}\)
\( \bigstar \) Distribute and simplify. (Assume all variables represent non-negative real numbers.)
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- Answers to Odd Exercises:
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129. \(3\sqrt{5}-5\\[6pt]\)
131. \(42 - 3 \sqrt { 21 }\\[6pt]\)
133. \(3 \sqrt { 2 } - 2 \sqrt { 3 }\)135. \(x + x \sqrt { y }\\[6pt]\)
137. \(2 a \sqrt { 7 b } - 4 b \sqrt { 5 a }\\[6pt]\)
139. \(\sqrt { 6 } + \sqrt { 14 } - \sqrt { 15 } - \sqrt { 35 }\)
141. \(18 \sqrt { 2 } + 2 \sqrt { 3 } - 12 \sqrt { 6 } - 4\)
143. \(8 - 2 \sqrt { 15 }\)
145. \(10\)
147. \(a - 2 \sqrt { 2 a b } + 2 b\)
\( \bigstar \) Divide and simplify. (Assume all variables represent non-negative real numbers.)
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- Answers to Odd Exercises:
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149. \(5\) 151. \(\dfrac { 2 \sqrt { 6 } } { 5 }\) 153. \(3 x ^ { 2 } \sqrt { 5 }\) 155. \( 4y \sqrt{2} \) 157. \(11 x y^4 \sqrt { x}\)
E: Rationalize the Denominator.
Exercise \(\PageIndex{5}\)
\( \bigstar \) Rationalize the denominator. (Assume all variables represent positive real numbers.)
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- Answers to Odd Exercises:
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161. \(\dfrac { \sqrt { 5 } } { 5 }\) 163. \(\dfrac { \sqrt { 6 } } { 3 }\) 165. \(\dfrac { \sqrt { 10 } } { 4 }\) 167. \(\dfrac { 3 - \sqrt { 15 } } { 3 }\) 169. \(\dfrac { \sqrt { 7 x } } { 7 x }\) 171. \(\dfrac { \sqrt { a b } } { 5 b }\)
\( \bigstar \) Rationalize the denominator. (Assume all variables represent positive real numbers.)
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- Answers to Odd Exercises:
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173. \(3\sqrt { 10 } + 9\) 175. \(\dfrac { \sqrt { 5 } - \sqrt { 3 } } { 2 }\) 177. \(- 1 + \sqrt { 2 }\) 179. \(\dfrac { - 5 - 3 \sqrt { 5 } } { 2} \) 181. \(- 4 - \sqrt { 15 }\) 183. \(\dfrac { 15 - 7 \sqrt { 6 } } { 23 }\)
\( \bigstar \) Rationalize the denominator. (Assume all variables represent positive real numbers.)
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- Answers to Odd Exercises:
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185. \(\sqrt { x } - \sqrt { y }\\[6pt]\)
187. \(\dfrac { x ^ { 2 } + 2 x \sqrt { y } + y } { x ^ { 2 } - y }\)189. \(\dfrac { a - 2 \sqrt { a b } + b } { a - b }\\[6pt]\)
191. \(\dfrac { 5 \sqrt { x } + 2 x } { 25 - 4 x }\)193. \(\dfrac { x \sqrt { 2 } + 3 \sqrt { x y } + y \sqrt { 2 } } { 2 x - y }\) 195. \(\dfrac { 2 x + 1 + \sqrt { 2 x + 1 } } { 2 x }\\[6pt]\)
197. \(x + \sqrt { x ^ { 2 } - 1 }\)
\( \bigstar \)