0.03e: Exercises - Square Roots

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A: Simplify Square Roots

Exercise \(\PageIndex{1}\)

\( \bigstar \) Simplify. (Assume all variable expressions represent positive numbers, so absolute values is not needed.)

\(\sqrt { 9 x ^ { 2 } }\\[6pt]\)
\(\sqrt { 16 y ^ { 2 } }\\[6pt]\)
\(\sqrt { 36 a ^ { 4 } }\\[6pt]\)
\(\sqrt { 100 a ^ { 8 } }\)

\(\sqrt { 4 a ^ { 6 } }\\[6pt]\)
\(\sqrt { a ^ { 10 } }\\[6pt]\)
\(\sqrt { 18 a ^ { 4 } b ^ { 5 } }\\[6pt]\)
\(\sqrt { 48 a ^ { 5 } b ^ { 3 } }\)

\(\sqrt { 49 a ^ { 2 } }\\[6pt]\)
\(\sqrt { 64 b ^ { 2 } }\\[6pt]\)
\(\sqrt { x ^ { 2 } y ^ { 2 } }\\[6pt]\)
\(\sqrt { 25 x ^ { 2 } y ^ { 2 } z ^ { 2 } }\)

\(\sqrt { 180 x ^ { 3 } }\\[6pt]\)
\(\sqrt { 150 y ^ { 3 } }\\[6pt]\)
\(\sqrt { 49 a ^ { 3 } b ^ { 2 } }\\[6pt]\)
\(\sqrt { 4 a ^ { 4 } b ^ { 3 } c }\)

\(\sqrt { 45 x ^ { 5 } y ^ { 3 } }\\[6pt]\)
\(\sqrt { 50 x ^ { 6 } y ^ { 4 } }\\[6pt]\)
\(\sqrt { 64 r ^ { 2 } s ^ { 6 } t ^ { 5 } }\\[6pt]\)
\(\sqrt { 144 r ^ { 8 } s ^ { 6 } t ^ { 2 } }\)

Answers to Odd Exercises:

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.)

\(\sqrt { ( 5 x - 4 ) ^ { 2 } }\\[6pt]\)
\(\sqrt { ( 3 x - 5 ) ^ { 4 } }\\[6pt]\)
\(\sqrt { x ^ { 2 } - 6 x + 9 }\\[6pt]\)
\(\sqrt { x ^ { 2 } - 10 x + 25 }\\[6pt]\)
\(\sqrt { 4 x ^ { 2 } + 12 x + 9 }\)

\(\sqrt { 9 x ^ { 2 } + 6 x + 1 }\\[6pt]\)
\(\sqrt { ( x + 1 ) ^ { 2 } }\\[6pt]\)
\(\sqrt { ( 2 x + 3 ) ^ { 2 } }\\[6pt]\)
\(\sqrt { 4 ( 3 x - 1 ) ^ { 2 } }\\[6pt]\)
\(\sqrt { 9 ( 2 x + 3 ) ^ { 2 } }\)

\(- 3 \sqrt { 4 x ^ { 2 } }\\[6pt]\)
\(7 \sqrt { 9 y ^ { 2 } }\\[6pt]\)
\(- 5 x \sqrt { 4 x ^ { 2 } y }\\[6pt]\)
\(- 3 y \sqrt { 16 x ^ { 3 } y ^ { 2 } }\\[6pt]\)
\(12 a b \sqrt { a ^ { 5 } b ^ { 3 } }\)

\(6 a ^ { 2 } b \sqrt { 9 a ^ { 7 } b ^ { 2 } }\)
\(\sqrt { \dfrac { 9 x ^ { 3 } } { 25 y ^ { 2 } } }\\[6pt]\)
\(\sqrt { \dfrac { 4 x ^ { 5 } } { 9 y ^ { 4 } } }\\[6pt]\)
\(\sqrt { \dfrac { m ^ { 7 } } { 36 n ^ { 4 } } }\)

\(\sqrt { \dfrac { 147 m ^ { 9 } } { n ^ { 6 } } }\\[6pt]\)
\(\sqrt { \dfrac { 2 r ^ { 2 } s ^ { 5 } } { 25 t ^ { 4 } } }\\[6pt]\)
\(\sqrt { \dfrac { 36 r ^ { 5 } } { s ^ { 2 } t ^ { 6 } } }\\[6pt]\)

Answers to Odd Exercises:

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

\(10 \sqrt { 3 } - 5 \sqrt { 3 }\\[6pt]\)
\(15 \sqrt { 6 } - 8 \sqrt { 6 }\\[6pt]\)
\(9 \sqrt { 3 } + 5 \sqrt { 3 }\\[6pt]\)
\(12 \sqrt { 6 } + 3 \sqrt { 6 }\\[6pt]\)
\(4 \sqrt { 5 } - 7 \sqrt { 5 } - 2 \sqrt { 5 }\)

\(3 \sqrt { 10 } - 8 \sqrt { 10 } - 2 \sqrt { 10 }\\[6pt]\)
\(\sqrt { 6 } - 4 \sqrt { 6 } + 2 \sqrt { 6 }\\[6pt]\)
\(5 \sqrt { 10 } - 15 \sqrt { 10 } - 2 \sqrt { 10 }\\[6pt]\)
\(13 \sqrt { 7 } - 6 \sqrt { 2 } - 5 \sqrt { 7 } + 5 \sqrt { 2 }\\[6pt]\)
\(10 \sqrt { 13 } - 12 \sqrt { 15 } + 5 \sqrt { 13 } - 18 \sqrt { 15 }\)

\(6 \sqrt { 5 } - ( 4 \sqrt { 3 } - 3 \sqrt { 5 } )\\[6pt]\)
\(- 12 \sqrt { 2 } - ( 6 \sqrt { 6 } + \sqrt { 2 } )\\[6pt]\)
\(( 2 \sqrt { 5 } - 3 \sqrt { 10 } ) - ( \sqrt { 10 } + 3 \sqrt { 5 } )\\[6pt]\)
\(( - 8 \sqrt { 3 } + 6 \sqrt { 15 } ) - ( \sqrt { 3 } - \sqrt { 15 } )\\[6pt]\)

Answers to Odd Exercises:

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

\(\sqrt { 75 } - \sqrt { 12 }\\[6pt]\)
\(\sqrt { 24 } - \sqrt { 54 }\\[6pt]\)
\(\sqrt { 32 } + \sqrt { 27 } - \sqrt { 8 }\\[6pt]\)
\(\sqrt { 20 } + \sqrt { 48 } - \sqrt { 45 }\\[6pt]\)
\(\sqrt { 28 } - \sqrt { 27 } + \sqrt { 63 } - \sqrt { 12 }\\[6pt]\)
\(\sqrt { 90 } + \sqrt { 24 } - \sqrt { 40 } - \sqrt { 54 }\)

\(\sqrt { 45 } - \sqrt { 80 } + \sqrt { 245 } - \sqrt { 5 }\\[6pt]\)
\(\sqrt { 108 } + \sqrt { 48 } - \sqrt { 75 } - \sqrt { 3 }\\[6pt]\)
\(4 \sqrt { 2 } - ( \sqrt { 27 } - \sqrt { 72 } )\\[6pt]\)
\(- 3 \sqrt { 5 } - ( \sqrt { 20 } - \sqrt { 50 } )\\[6pt]\)
\(2 \sqrt { 27 } - 2 \sqrt { 12 }\\[6pt]\)
\(3 \sqrt { 50 } - 4 \sqrt { 32 }\)

\(3 \sqrt { 243 } - 2 \sqrt { 18 } - \sqrt { 48 }\\[6pt]\)
\(6 \sqrt { 216 } - 2 \sqrt { 24 } - 2 \sqrt { 96 }\\[6pt]\)
\(2 \sqrt { 18 } - 3 \sqrt { 75 } - 2 \sqrt { 98 } + 4 \sqrt { 48 }\\[6pt]\)
\(2 \sqrt { 45 } - \sqrt { 12 } + 2 \sqrt { 20 } - \sqrt { 108 }\\[6pt]\)
\(( 2 \sqrt { 363 } - 3 \sqrt { 96 } ) - ( 7 \sqrt { 12 } - 2 \sqrt { 54 } )\\[6pt]\)
\(( 2 \sqrt { 288 } + 3 \sqrt { 360 } ) - ( 2 \sqrt { 72 } - 7 \sqrt { 40 } )\)

Answers to Odd Exercises:

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.)

\(\sqrt { 2 x } - 4 \sqrt { 2 x }\\[6pt]\)
\(5 \sqrt { 3 y } - 6 \sqrt { 3 y }\\[6pt]\)
\(9 \sqrt { x } + 7 \sqrt { x }\\[6pt]\)
\(- 8 \sqrt { y } + 4 \sqrt { y }\)

\(7 x \sqrt { y } - 3 x \sqrt { y } + x \sqrt { y }\\[6pt]\)
\(10 y ^ { 2 } \sqrt { x } - 12 y ^ { 2 } \sqrt { x } - 2 y ^ { 2 } \sqrt { x }\\[6pt]\)
\(2 \sqrt { a b } - 5 \sqrt { a } + 6 \sqrt { a b } - 10 \sqrt { a }\\[6pt]\)
\(- 3 x \sqrt { y } + 6 \sqrt { y } - 4 x \sqrt { y } - 7 \sqrt { y }\)

\(5 \sqrt { x y } - ( 3 \sqrt { x y } - 7 \sqrt { x y } )\\[6pt]\)
\(- 8 a \sqrt { b } - ( 2 a \sqrt { b } - 4 \sqrt { a b } )\\[6pt]\)
\(( 3 \sqrt { 2 x } - \sqrt { 3 x } ) - ( \sqrt { 2 x } - 7 \sqrt { 3 x } )\\[6pt]\)
\(( \sqrt { y } - 4 \sqrt { 2 y } ) - ( \sqrt { y } - 5 \sqrt { 2 y } )\)

Answers to Odd Exercises:

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.)

\(\sqrt { 81 b } + \sqrt { 4 b }\\[6pt]\)
\(\sqrt { 100 a } + \sqrt { a }\\[6pt]\)
\(\sqrt { 9 a ^ { 2 } b } - \sqrt { 36 a ^ { 2 } b }\\[6pt]\)
\(\sqrt { 50 a ^ { 2 } } - \sqrt { 18 a ^ { 2 } }\\[6pt]\)
\(\sqrt { 49 x } - \sqrt { 9 y } + \sqrt { x } - \sqrt { 4 y }\\[6pt]\)
\(\sqrt { 9 x } + \sqrt { 64 y } - \sqrt { 25 x } - \sqrt { y }\)

\(7 \sqrt { 8 x } - ( 3 \sqrt { 16 y } - 2 \sqrt { 18 x } )\\[6pt]\)
\(2 \sqrt { 64 y } - ( 3 \sqrt { 32 y } - \sqrt { 81 y } )\\[6pt]\)
\(2 \sqrt { 9 m ^ { 2 } n } - 5 m \sqrt { 9 n } + \sqrt { m ^ { 2 } n }\\[6pt]\)
\(4 \sqrt { 18 n ^ { 2 } m } - 2 n \sqrt { 8 m } + n \sqrt { 2 m }\\[6pt]\)
\(\sqrt { 4 x ^ { 2 } y } - \sqrt { 9 x y ^ { 2 } } - \sqrt { 16 x ^ { 2 } y } + \sqrt { y ^ { 2 } x }\\[6pt]\)
\(\sqrt { 32 x ^ { 2 } y ^ { 2 } } + \sqrt { 12 x ^ { 2 } y } - \\ \sqrt { 18 x ^ { 2 } y ^ { 2 } } - \sqrt { 27 x ^ { 2 } y }\)

\(\left( \sqrt { 9 x ^ { 2 } y } - \sqrt { 16 y } \right) - \left( \sqrt { 49 x ^ { 2 } y } - 4 \sqrt { y } \right)\\[6pt]\)
\(\left( \sqrt { 72 x ^ { 2 } y ^ { 2 } } - \sqrt { 18 x ^ { 2 } y } \right) \\ - \left( \sqrt { 50 x ^ { 2 } y ^ { 2 } } + x \sqrt { 2 y } \right)\\[6pt]\)
\(\sqrt { 12 m ^ { 4 } n } - m \sqrt { 75 m ^ { 2 } n } + 2 \sqrt { 27 m ^ { 4 } n }\\[6pt]\)
\(5 n \sqrt { 27 m n ^ { 2 } } + 2 \sqrt { 12 m n ^ { 4 } } - n \sqrt { 3 m n ^ { 2 } }\\[6pt]\)
\(2 \sqrt { 27 a ^ { 3 } b } - a \sqrt { 48 a b } - a \sqrt { 144 a ^ { 3 } b }\\[6pt]\)
\(2 \sqrt { 98 a ^ { 4 } b } - 2 a \sqrt { 162 a ^ { 2 } b } + a \sqrt { 200 b }\)

Answers to Odd Exercises:

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.)

\(\sqrt { 3 } \cdot \sqrt { 7 }\\[6pt]\)
\(\sqrt { 2 } \cdot \sqrt { 5 }\\[6pt]\)
\(\sqrt { 6 } \cdot \sqrt { 12 }\\[6pt]\)
\(\sqrt { 10 } \cdot \sqrt { 15 }\\[6pt]\)
\(\sqrt { 2 } \cdot \sqrt { 6 }\)

\(\sqrt { 5 } \cdot \sqrt { 15 }\\[6pt]\)
\(\sqrt { 7 } \cdot \sqrt { 7 }\\[6pt]\)
\(\sqrt { 12 } \cdot \sqrt { 12 }\\[6pt]\)
\(2 \sqrt { 5 } \cdot 7 \sqrt { 10 }\\[6pt]\)
\(3 \sqrt { 15 } \cdot 2 \sqrt { 6 }\)

\(( 2 \sqrt { 5 } ) ^ { 2 }\\[6pt]\)
\(( 6 \sqrt { 2 } ) ^ { 2 }\\[6pt]\)
\(\sqrt { 2 x } \cdot \sqrt { 2 x }\\[6pt]\)
\(\sqrt { 5 y } \cdot \sqrt { 5 y }\\[6pt]\)

\(\sqrt { 3 a } \cdot \sqrt { 12 }\\[6pt]\)
\(\sqrt { 3 a } \cdot \sqrt { 2 a }\\[6pt]\)
\(4 \sqrt { 2 x } \cdot 3 \sqrt { 6 x }\\[6pt]\)
\(5 \sqrt { 10 y } \cdot 2 \sqrt { 2 y }\\[6pt]\)

Answers to Odd Exercises:

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.)

\(\sqrt { 5 } ( 3 - \sqrt { 5 } )\\[6pt]\)
\(\sqrt { 2 } ( \sqrt { 3 } - \sqrt { 2 } )\\[6pt]\)
\(3 \sqrt { 7 } ( 2 \sqrt { 7 } - \sqrt { 3 } )\\[6pt]\)
\(2 \sqrt { 5 } ( 6 - 3 \sqrt { 10 } )\\[6pt]\)
\(\sqrt { 6 } ( \sqrt { 3 } - \sqrt { 2 } )\)

\(\sqrt { 15 } ( \sqrt { 5 } + \sqrt { 3 } )\\[6pt]\)
\(\sqrt { x } ( \sqrt { x } + \sqrt { x y } )\\[6pt]\)
\(\sqrt { y } ( \sqrt { x y } + \sqrt { y } )\\[6pt]\)
\(\sqrt { 2 a b } ( \sqrt { 14 a } - 2 \sqrt { 10 b } )\\[6pt]\)
\(\sqrt { 6 a b } ( 5 \sqrt { 2 a } - \sqrt { 3 b } )\)

\(( \sqrt { 2 } - \sqrt { 5 } ) ( \sqrt { 3 } + \sqrt { 7 } )\\[6pt]\)
\(( \sqrt { 3 } + \sqrt { 2 } ) ( \sqrt { 5 } - \sqrt { 7 } )\\[6pt]\)
\(( 2 \sqrt { 3 } - 4 ) ( 3 \sqrt { 6 } + 1 )\\[6pt]\)
\(( 5 - 2 \sqrt { 6 } ) ( 7 - 2 \sqrt { 3 } )\\[6pt]\)
\(( \sqrt { 5 } - \sqrt { 3 } ) ^ { 2 }\)

\(( \sqrt { 7 } - \sqrt { 2 } ) ^ { 2 }\\[6pt]\)
\(( 2 \sqrt { 3 } + \sqrt { 2 } ) ( 2 \sqrt { 3 } - \sqrt { 2 } )\\[6pt]\)
\(( \sqrt { 2 } + 3 \sqrt { 7 } ) ( \sqrt { 2 } - 3 \sqrt { 7 } )\\[6pt]\)
\(( \sqrt { a } - \sqrt { 2 b } ) ^ { 2 }\\[6pt]\)
\(( \sqrt { a b } + 1 ) ^ { 2 }\)

Answers to Odd Exercises:

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.)

\(\dfrac { \sqrt { 75 } } { \sqrt { 3 } }\\[6pt]\)
\(\dfrac { \sqrt { 360 } } { \sqrt { 10 } }\)

\(\dfrac { \sqrt { 72 } } { \sqrt { 75 } } \\[6pt]\)
\(\dfrac { \sqrt { 90 } } { \sqrt { 98 } }\)

\(\dfrac { \sqrt { 90 x ^ { 5 } } } { \sqrt { 2 x } }\\[6pt]\)
\(\dfrac { \sqrt { 96 y ^ { 3 } } } { \sqrt { 3 y } }\)

\(\dfrac { \sqrt { 96 y ^ { 3 } } } { \sqrt { 3 y } }\\[6pt]\)
\(\dfrac { \sqrt { 162 x ^ { 7 } y ^ { 5 } } } { \sqrt { 2 x y } }\)

\(\dfrac { \sqrt { 363 x ^ { 4 } y ^ { 9 } } } { \sqrt { 3 x y } }\)

Answers to Odd Exercises:

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.)

\(\dfrac { 1 } { \sqrt { 5 } }\\[6pt]\)
\(\dfrac { 1 } { \sqrt { 6 } }\\[6pt]\)

\(\dfrac { \sqrt { 2 } } { \sqrt { 3 } }\\[6pt]\)
\(\dfrac { \sqrt { 3 } } { \sqrt { 7 } }\\[6pt]\)

\(\dfrac { 5 } { 2 \sqrt { 10 } }\\[6pt]\)
\(\dfrac { 3 } { 5 \sqrt { 6 } }\\[6pt]\)

\(\dfrac { \sqrt { 3 } - \sqrt { 5 } } { \sqrt { 3 } }\\[6pt]\)
\(\dfrac { \sqrt { 6 } - \sqrt { 2 } } { \sqrt { 2 } }\\[6pt]\)

\(\dfrac { 1 } { \sqrt { 7 x } }\\[6pt]\)
\(\dfrac { 1 } { \sqrt { 3 y } }\\[6pt]\)

\(\dfrac { a } { 5 \sqrt { a b } }\\[6pt]\)
\(\dfrac { 3 b ^ { 2 } } { 2 \sqrt { 3 a b } }\\[6pt]\)

Answers to Odd Exercises:

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.)

\(\dfrac { 3 } { \sqrt { 10 } - 3 }\\[6pt]\)
\(\dfrac { 2 } { \sqrt { 6 } - 2 }\)

\(\dfrac { 1 } { \sqrt { 5 } + \sqrt { 3 } }\\[6pt]\)
\(\dfrac { 1 } { \sqrt { 7 } - \sqrt { 2 } }\)

\(\dfrac { \sqrt { 3 } } { \sqrt { 3 } + \sqrt { 6 } }\\[6pt]\)
\(\dfrac { \sqrt { 5 } } { \sqrt { 5 } + \sqrt { 15 } }\)

\(\dfrac { 10 } { 5 - 3 \sqrt { 5 } }\\[6pt]\)
\(\dfrac { - 2 \sqrt { 2 } } { 4 - 3 \sqrt { 2 } }\)

\(\dfrac { \sqrt { 3 } + \sqrt { 5 } } { \sqrt { 3 } - \sqrt { 5 } }\\[6pt]\)
\(\dfrac { \sqrt { 10 } - \sqrt { 2 } } { \sqrt { 10 } + \sqrt { 2 } }\)

\(\dfrac { 2 \sqrt { 3 } - 3 \sqrt { 2 } } { 4 \sqrt { 3 } + \sqrt { 2 } }\\[6pt]\)
\(\dfrac { 6 \sqrt { 5 } + 2 } { 2 \sqrt { 5 } - \sqrt { 2 } }\)

Answers to Odd Exercises:

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.)

\(\dfrac { x - y } { \sqrt { x } + \sqrt { y } }\\[6pt]\)
\(\dfrac { x - y } { \sqrt { x } - \sqrt { y } }\\[6pt]\)
\(\dfrac { x + \sqrt { y } } { x - \sqrt { y } }\\[6pt]\)
\(\dfrac { x - \sqrt { y } } { x + \sqrt { y } }\)

\(\dfrac { \sqrt { a } - \sqrt { b } } { \sqrt { a } + \sqrt { b } }\\[6pt]\)
\(\dfrac { \sqrt { a b } + \sqrt { 2 } } { \sqrt { a b } - \sqrt { 2 } }\\[6pt]\)
\(\dfrac { \sqrt { x } } { 5 - 2 \sqrt { x } }\\[6pt]\)
\(\dfrac { 1 } { \sqrt { x } - y }\)

\(\dfrac { \sqrt { x } + \sqrt { 2 y } } { \sqrt { 2 x } - \sqrt { y } }\\[6pt]\)
\(\dfrac { \sqrt { 3 x } - \sqrt { y } } { \sqrt { x } + \sqrt { 3 y } }\\[6pt]\)
\(\dfrac { \sqrt { 2 x + 1 } } { \sqrt { 2 x + 1 } - 1 }\\[6pt]\)
\(\dfrac { \sqrt { x + 1 } } { 1 - \sqrt { x + 1 } }\)

\(\dfrac { \sqrt { x + 1 } + \sqrt { x - 1 } } { \sqrt { x + 1 } - \sqrt { x - 1 } }\\[6pt]\)
\(\dfrac { \sqrt { 2 x + 3 } - \sqrt { 2 x - 3 } } { \sqrt { 2 x + 3 } + \sqrt { 2 x - 3 } }\)

Answers to Odd Exercises:

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. \(x + \sqrt { x ^ { 2 } - 1 }\)

\( \bigstar \)

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