0.04e: Exercises - Rational Exponents

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A: Radical to Exponential Notation

Exercise \(\PageIndex{1}\)

\( \bigstar \) Express using rational exponents.

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

\(\sqrt [ 3 ] { 3 } \\[5pt]\)
\(\sqrt [ 4 ] { 5 } \\[5pt]\)

\(\sqrt [ 3 ] { 5 ^ { 2 } } \\[5pt]\)
\(\sqrt [ 4 ] { 2 ^ { 3 } } \\[5pt]\)

\(\sqrt [ 3 ] { 49 } \\[5pt]\)
\(\sqrt [ 3 ] { 9 } \\[5pt]\)

\(\sqrt [ 5 ] { x } \\[5pt]\)
\(\sqrt [ 6 ] { x } \\[5pt]\)

\(\sqrt [ 6 ] { x ^ { 7 } } \\[5pt]\)
\(\sqrt [ 5 ] { x ^ { 4 } } \\[5pt]\)

\(\dfrac { 1 } { \sqrt { x } } \\[5pt]\)
\(\dfrac { 1 } { \sqrt [ 3 ] { x ^ { 2 } } } \\[5pt]\)

Answers to odd exercises:

1. \(10 ^ { 1 / 2 } \)

3. \(3 ^ { 1 / 3 } \)

5. \(5 ^ { 2 / 3 } \)

7. \(7 ^ { 2 / 3 } \)

9. \(x ^ { 1 / 5 } \)

11. \(x ^ { 7 / 6 } \)

13. \(x ^ { - 1 / 2 }\)

B: Exponential to Radical Notation.

Exercise \(\PageIndex{2}\)

\( \bigstar \) Express in radical form.

\(10 ^ { 1 / 2 } \\[5pt]\)
\(11 ^ { 1 / 3 } \)

\(7 ^ { 2 / 3 } \\[5pt]\)
\(2 ^ { 3 / 5 } \)

\(x ^ { 3 / 4 } \\[5pt]\)
\(x ^ { 5 / 6 } \)

\(x ^ { - 1 / 2 } \\[5pt]\)
\(x ^ { - 3 / 4 } \)

\(\left( \frac { 1 } { x } \right) ^ { - 1 / 3 } \\[5pt]\)
\(\left( \frac { 1 } { x } \right) ^ { - 3 / 5 } \)

\(( 2 x + 1 ) ^ { 2 / 3 } \\[5pt]\)
\(( 5 x - 1 ) ^ { 1 / 2 } \)

Answers to odd exercises:

15. \(\sqrt { 10 }\)

17. \(\sqrt [ 3 ] { 49 } \)

19. \(\sqrt [ 4 ] { x ^ { 3 } }\)

21. \(\dfrac { 1 } { \sqrt { x } } \)

23. \(\sqrt [ 3 ] { x } \)

25. \(\sqrt [ 3 ] { ( 2 x + 1 ) ^ { 2 } } \)

C: Exponential to Radical Form; then Simplify.

Exercise \(\PageIndex{3}\)

\( \bigstar \) Write as a radical and then simplify.

\(64 ^ { 1 / 2 } \\[5pt]\)
\(49 ^ { 1 / 2 } \\[5pt]\)
\(\left( \dfrac { 1 } { 4 } \right) ^ { 1 / 2 } \\[5pt]\)
\(\left( \dfrac { 4 } { 9 } \right) ^ { 1 / 2 } \)

\(4 ^ { - 1 / 2 } \\[2pt]\)
\(9 ^ { - 1 / 2 } \\[2pt]\)
\(\left( \dfrac { 1 } { 4 } \right) ^ { - 1 / 2 } \\[5pt]\)
\(\left( \dfrac { 1 } { 16 } \right) ^ { - 1 / 2 } \)

\(8 ^ { 1 / 3 } \\[2pt] \)
\(125 ^ { 1 / 3 } \\[2pt]\)
\(\left( \dfrac { 1 } { 27 } \right) ^ { 1 / 3 } \\[5pt]\)
\(\left( \dfrac { 8 } { 125 } \right) ^ { 1 / 3 } \\[5pt]\)
\(( - 27 ) ^ { 1 / 3 } \)

\(( - 64 ) ^ { 1 / 3 } \\[5pt]\)
\(16 ^ { 1 / 4 } \\[5pt]\)
\(625 ^ { 1 / 4 } \\[5pt]\)
\(81 ^ { - 1 / 4 } \\[5pt]\)
\(16 ^ { - 1 / 4 } \)

\(100,000 ^ { 1 / 5 } \\[5pt]\)
\(( - 32 ) ^ { 1 / 5 } \\[5pt]\)
\(\left( \dfrac { 1 } { 32 } \right) ^ { 1 / 5 } \\[5pt]\)
\(\left( \dfrac { 1 } { 243 } \right) ^ { 1 / 5 } \)

Answers to odd exercises:

27. \(8\)
29. \(\dfrac{1}{2} \)

31. \(\dfrac{1}{2} \\[5pt]\)
33. \(2 \)

35. \(2\)
37. \( \dfrac{1}{3} \)

39. \(-3 \\[5pt]\)
41. \(2 \)

43. \(\dfrac{1}{3} \\[5pt]\)
45. \( 10 \)

47. \(\dfrac{1}{2} \)

\( \bigstar \) Write as a radical and then simplify.

\(9 ^ { 3 / 2 } \\[5pt]\)
\(4 ^ { 3 / 2 } \)

\(8 ^ { 5 / 3 } \\[5pt]\)
\(27 ^ { 2 / 3 } \)

\(16 ^ { 3 / 2 } \\[5pt]\)
\(32 ^ { 2 / 5 } \)

\(\left( \dfrac { 1 } { 16 } \right) ^ { 3 / 4 } \\[5pt]\)
\(\left( \dfrac { 1 } { 81 } \right) ^ { 3 / 4 } \)

\(( - 27 ) ^ { 2 / 3 } \\[5pt]\)
\( ( - 27 ) ^ { 4 / 3 } \)

\(( - 32 ) ^ { 3 / 5 } \\[5pt]\)
\(( - 32 ) ^ { 4 / 5 } \)

Answers to odd exercises:

49. \(27 \)

51. \(32\)

53. \(64 \)

55. \(\dfrac{1}{8} \)

57. \(9 \)

59. \(-8\)

D: Exponential Operations. PRODUCTS and POWERS of Products

Exercise \(\PageIndex{4}\)

\( \bigstar \) Perform the operations and simplify. Leave answers in exponential form.

\(5 ^ { 3 / 2 } \cdot 5 ^ { 1 / 2 } \\[5pt]\)
\(3 ^ { 2 / 3 } \cdot 3 ^ { 7 / 3 } \\[5pt]\)
\(5 ^ { 1 / 2 } \cdot 5 ^ { 1 / 3 } \\[5pt]\)
\(2 ^ { 1 / 6 } \cdot 2 ^ { 3 / 4 } \)

\(y ^ { 1 / 4 } \cdot y ^ { 2 / 5 } \\[5pt]\)
\(x ^ { 1 / 2 } \cdot x ^ { 1 / 4 } \\[5pt]\)
\((u^{12}v^{18})^{\tfrac{1}{6}} \\[5pt]\)
\((r^{9}s^{12})^{\tfrac{1}{3}} \)

\(\left( 8 ^ { 1 / 2 } \right) ^ { 2 / 3 } \\[5pt]\)
\(\left( 3 ^ { 6 } \right) ^ { 2 / 3 } \\[5pt]\)
\(\left( x ^ { 2 / 3 } \right) ^ { 1 / 2 } \\[5pt]\)
\(\left( y ^ { 3 / 4 } \right) ^ { 4 / 5 } \)

\(\left( y ^ { 8 } \right) ^ { - 1 / 2 } \\[5pt]\)
\(\left( y ^ { 6 } \right) ^ { - 2 / 3 } \\[5pt]\)
\(\left( 4 x ^ { 2 } y ^ { 4 } \right) ^ { 1 / 2 } \\[5pt]\)
\(\left( 9 x ^ { 6 } y ^ { 2 } \right) ^ { 1 / 2 } \)

\(\left( 2 x ^ { 1 / 3 } y ^ { 2 / 3 } \right) ^ { 3 } \\[5pt]\)
\(\left( 8 x ^ { 3 / 2 } y ^ { 1 / 2 } \right) ^ { 2 } \\[5pt]\)
\(\left( 36 x ^ { 4 } y ^ { 2 } \right) ^ { - 1 / 2 } \\[5pt]\)
\(\left( 8 x ^ { 3 } y ^ { 6 } z ^ { - 3 } \right) ^ { - 1 / 3 } \)

Answers to odd exercises:

61. \(25 \\[5pt]\)
63. \(5 ^ { 5 / 6 } \)

65. \(y ^ { 13 / 20 } \\[5pt]\)
67.\(u^{2}v^{3} \)

69. \(2 \\[5pt]\)
71. \(x ^ { 1 / 3 } \)

73. \(\dfrac { 1 } { y ^ { 4 } } \\[5pt]\)
75. \(2 x y ^ { 2 }\)

77. \(8 x y ^ { 2 } \\[5pt]\)
79. \(\dfrac { 1 } { 6 x ^ { 2 } y } \)

\( \bigstar \) Perform the operations and simplify. Leave answers in exponential form.

\(\left(27 q^{\tfrac{3}{2}}\right)^{\tfrac{4}{3}} \\[5pt]\)
\(\left(64 s^{\tfrac{3}{7}}\right)^{\tfrac{1}{6}} \\[5pt]\)
\(\left(a^{\tfrac{1}{3}} b^{\tfrac{2}{3}}\right)^{\tfrac{3}{2}} \)

\( \left( m^{\tfrac{4}{3}} n^{\tfrac{1}{2}}\right)^{\tfrac{3}{4}} \\[5pt]\)
\(\left(16 u^{\tfrac{1}{3}}\right)^{\tfrac{3}{4}} \\[5pt]\)
\(\left(625 n^{\tfrac{8}{3}}\right)^{\tfrac{3}{4}} \)

\(\left(4 p^{\tfrac{1}{3}} q^{\tfrac{1}{2}}\right)^{\tfrac{3}{2}} \\[5pt]\)
\(\left(9 x^{\tfrac{2}{5}} y^{\tfrac{3}{5}}\right)^{\tfrac{5}{2}} \\[5pt]\)
\(\left( 16 x ^ { 2 } y ^ { - 1 / 3 } z ^ { 2 / 3 } \right) ^ { - 3 / 2 } \)

\(\left( 81 x ^ { 8 } y ^ { - 4 / 3 } z ^ { - 4 } \right) ^ { - 3 / 4 } \\[5pt]\)
\(\left( 100 a ^ { - 2 / 3 } b ^ { 4 } c ^ { - 3 / 2 } \right) ^ { - 1 / 2 } \\[5pt]\)
\(\left( 125 a ^ { 9 } b ^ { - 3 / 4 } c ^ { - 1 } \right) ^ { - 1 / 3 } \)

Answers to odd exercises:

81. \(81 q^{2} \\[2pt]\)
83. \(a^{\tfrac{1}{2}} b\)

85. \(8 u^{\tfrac{1}{4}} \\[5pt]\)
87. \(8 p^{\tfrac{1}{2}} q^{\tfrac{3}{4}} \)

85. \(8 u^{\tfrac{1}{4}} \\[5pt]\)
87. \(8 p^{\tfrac{1}{2}} q^{\tfrac{3}{4}}\)

89. \(\dfrac { y ^ { 1 / 2 } } { 64 x ^ { 3 } z } \\[5pt]\)
91. \(\dfrac { a ^ { 1 / 3 } b ^ { 3 / 4 } } { 10 b ^ { 2 } }\)

E: Exponential Operations. QUOTIENTS and POWERS of Quotients

Exercise \(\PageIndex{5}\)

\( \bigstar \) Perform the operations and simplify. Leave answers in exponential form.

\(\dfrac { 5 ^ { 11 / 3 } } { 5 ^ { 2 / 3 } } \\[5pt]\)
\(\dfrac { 2 ^ { 9 / 2 } } { 2 ^ { 1 / 2 } } \\[5pt]\)
\(\dfrac { 2 a ^ { 2 / 3 } } { a ^ { 1 / 6 } } \\[5pt]\)
\(\dfrac { 3 b ^ { 1 / 2 } } { b ^ { 1 / 3 } } \)

\(\dfrac{r^{\tfrac{5}{2}} \cdot r^{-\tfrac{1}{2}}}{r^{-\tfrac{3}{2}}} \\[5pt]\)
\(\dfrac{a^{\tfrac{3}{4}} \cdot a^{-\tfrac{1}{4}}}{a^{-\tfrac{10}{4}}} \\[5pt]\)
\(\dfrac{c^{\tfrac{5}{3}} \cdot c^{-\tfrac{1}{3}}}{c^{-\tfrac{2}{3}}} \)

\(\dfrac{m^{\tfrac{7}{4}} \cdot m^{-\tfrac{5}{4}}}{m^{-\tfrac{2}{4}}} \\[16pt]\)
113 \(\dfrac { y ^ { 1 / 2 } y ^ { 2 / 3 } } { y ^ { 1 / 6 } } \\[16pt]\)
\(\dfrac { x ^ { 2 / 5 } x ^ { 1 / 2 } } { x ^ { 1 / 10 } } \)

\(\dfrac { x y } { x ^ { 1 / 2 } y ^ { 1 / 3 } } \)
\(\dfrac { x ^ { 5 / 4 } y } { x y ^ { 2 / 5 } } \)
\(\dfrac { 49 a ^ {5/7 } b ^ { 3 / 2 } } { 7 a ^ { 3 /7 } b ^ { 1 / 4 } } \\[5pt]\)
\(\dfrac { 16 a ^ { 5 / 6 } b ^ { 5 / 4 } } { 8 a ^ { 1 / 2 } b ^ { 2 / 3 } } \)

Answers to odd exercises:

101. \(125 \)

103. \(2 a ^ { 1 / 2 } \)

105. \(r^{\frac{7}{2}}\)

107. \(c^{2} \)

109. \(y\)

111. \(x ^ { 1 / 2 } y ^ { 2 / 3 } \)

113. \(7 a ^ { 2/7 } b ^ { 5 / 4 } \)

\( \bigstar \) Perform the operations and simplify. Leave answers in exponential form.

\(\left( \dfrac { a ^ { 3 / 4 } } { a ^ { 1 / 2 } } \right) ^ { 4 / 3 } \\[6pt] \)
\(\left( \dfrac { b ^ { 4 / 5 } } { b ^ { 1 / 10 } } \right) ^ { 10 / 3 } \\[6pt]\)
\(\left( \dfrac { 4 x ^ { 2 / 3 } } { y ^ { 4 } } \right) ^ { 1 / 2 } \\[6pt] \)
\(\left( \dfrac { 27 x ^ { 3 / 4 } } { y ^ { 9 } } \right) ^ { 1 / 3 } \)

\(\left( \dfrac { 27 x ^ { 3 / 4 } } { y ^ { 9 } } \right) ^ { 1 / 3 } \\[2pt] \)
\(\left(\dfrac{36 s^{\tfrac{1}{5}} t^{-\tfrac{3}{2}}}{s^{-\tfrac{9}{5}} t^{\tfrac{1}{2}}}\right)^{\tfrac{1}{2}} \\[2pt]\)
\(\left(\dfrac{27 b^{\tfrac{2}{3}} c^{-\tfrac{5}{2}}}{b^{-\tfrac{7}{3}} c^{\tfrac{1}{2}}}\right)^{\tfrac{1}{3}} \)

\(\left(\dfrac{8 x^{\tfrac{5}{3}} y^{-\tfrac{1}{2}}}{27 x^{-\tfrac{4}{3}} y^{\tfrac{5}{2}}}\right)^{\tfrac{1}{3}} \\[1pt]\)
\(\left(\dfrac{16 m^{\tfrac{1}{5}} n^{\tfrac{3}{2}}}{81 m^{\tfrac{9}{5}} n^{-\tfrac{1}{2}}}\right)^{\tfrac{1}{4}} \\[6pt] \)
\(\dfrac { \left( 9 x ^ { 2 / 3 } y ^ { 6 } \right) ^ { 3 / 2 } } { x ^ { 1 / 2 } y } \)

\(\dfrac { \left( 125 x ^ { 3 } y ^ { 3 / 5 } \right) ^ { 2 / 3 } } { x y ^ { 1 / 3 } } \\[16pt]\)
\(\dfrac { \left( 27 a ^ { 1 / 4 } b ^ { 3 / 2 } \right) ^ { 2 / 3 } } { a ^ { 1 / 6 } b ^ { 1 / 2 } } \\[16pt] \)
\(\dfrac { \left( 25 a ^ { 2 / 3 } b ^ { 4 / 3 } \right) ^ { 3 / 2 } } { a ^ { 1 / 6 } b ^ { 1 / 3 } } \)

Answers to odd exercises:

115. \(a ^ { 1 / 3 } \)

117. \(\dfrac { 2 x ^ { 1 / 3 } } { y ^ { 2 } } \)

119. \(\dfrac{6 s}{t} \)

121. \(\dfrac{2x}{3y} \)

123. \(27 x ^ { 1 / 2 } y ^ { 8 } \)

125. \(9 b ^ { 1 / 2 } \)

F: Radical to Exponential Form Operations.

Exercise \(\PageIndex{6} \)

\( \bigstar \) Rewrite in exponential form and then perform the operations.

\(\sqrt [ 3 ] { 9 } \cdot \sqrt [ 5 ] { 3 } \\[5pt]\)
\(\sqrt { 5 } \cdot \sqrt [ 5 ] { 25 } \\[5pt]\)
\(\sqrt { x } \cdot \sqrt [ 3 ] { x } \)

\(\sqrt { y } \cdot \sqrt [ 4 ] { y } \\[5pt]\)
\(\sqrt [ 3 ] { x ^ { 2 } } \cdot \sqrt [ 4 ] { x } \\[5pt]\)
\(\sqrt [ 5 ] { x ^ { 3 } } \cdot \sqrt [ 3 ] { x } \)

\(\dfrac { \sqrt [ 3 ] { 100 } } { \sqrt { 10 } } \\[5pt]\)
\(\dfrac { \sqrt [ 5 ] { 16 } } { \sqrt [ 3 ] { 4 } } \)

\(\dfrac { \sqrt [ 3 ] { a ^ { 2 } } } { \sqrt { a } } \\[5pt]\)
\(\dfrac { \sqrt [ 5 ] { b ^ { 4 } } } { \sqrt [ 3 ] { b } } \)

\(\dfrac { \sqrt [ 3 ] { x ^ { 2 } } } { \sqrt [ 5 ] { x ^ { 3 } } } \\[5pt]\)
\(\dfrac { \sqrt [ 4 ] { x ^ { 3 } } } { \sqrt [ 3 ] { x ^ { 2 } } } \)

\(\sqrt { \sqrt [ 5 ] { 16 } } \\[5pt]\)
\(\sqrt { \sqrt [ 3 ] { 9 } } \\[5pt]\)
\(\sqrt [ 3 ] { \sqrt [ 5 ] { 2 } } \)

Answers to odd exercises:

131. \(\sqrt [ 15 ] { 3 ^ { 13 } } \\[5pt]\)
133. \(\sqrt [ 6 ] { x ^ { 5 } } \)

135. \(\sqrt [ 12 ] { x ^ { 11 } } \\[5pt]\)
137. \(\sqrt [ 6 ] { 10 }\)

139. \(\sqrt [ 6 ] { a } \\[5pt]\)
141. \(\sqrt [ 15 ] { x }\)

143. \(\sqrt [ 5 ] { 4 } \\[5pt]\)
145. \(\sqrt [ 15 ] { 2 } )

.\( \bigstar \)

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