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Mathematics LibreTexts

0.4e: Exercises - Rational Exponents

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    38226
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    A: Radical to Exponential Notation

    Exercise \(\PageIndex{1}\) 

    \( \bigstar \) Express using rational exponents.

    1. \(\sqrt{10} \\[5pt]\)
    2. \(\sqrt{6} \\[5pt]\)
    1. \(\sqrt [ 3 ] { 3 } \\[5pt]\)
    2. \(\sqrt [ 4 ] { 5 } \\[5pt]\)
    1. \(\sqrt [ 3 ] { 5 ^ { 2 } } \\[5pt]\)
    2. \(\sqrt [ 4 ] { 2 ^ { 3 } } \\[5pt]\)
    1. \(\sqrt [ 3 ] { 49 } \\[5pt]\)
    2. \(\sqrt [ 3 ] { 9 } \\[5pt]\)
    1. \(\sqrt [ 5 ] { x } \\[5pt]\)
    2. \(\sqrt [ 6 ] { x } \\[5pt]\)
    1. \(\sqrt [ 6 ] { x ^ { 7 } } \\[5pt]\)
    2. \(\sqrt [ 5 ] { x ^ { 4 } } \\[5pt]\)
    1. \(\dfrac { 1 } { \sqrt { x } } \\[5pt]\)
    2. \(\dfrac { 1 } { \sqrt [ 3 ] { x ^ { 2 } } } \\[5pt]\)
    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.

    1. \(10 ^ { 1 / 2 } \\[5pt]\)
    2. \(11 ^ { 1 / 3 } \)
    1. \(7 ^ { 2 / 3 } \\[5pt]\)
    2. \(2 ^ { 3 / 5 } \)
    1. \(x ^ { 3 / 4 } \\[5pt]\)
    2. \(x ^ { 5 / 6 } \)
    1. \(x ^ { - 1 / 2 } \\[5pt]\)
    2. \(x ^ { - 3 / 4 } \)
    1. \(\left( \frac { 1 } { x } \right) ^ { - 1 / 3 } \\[5pt]\)
    2. \(\left( \frac { 1 } { x } \right) ^ { - 3 / 5 } \)
    1. \(( 2 x + 1 ) ^ { 2 / 3 } \\[5pt]\)
    2. \(( 5 x - 1 ) ^ { 1 / 2 } \)
    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.

    1. \(64 ^ { 1 / 2 } \\[5pt]\)
    2. \(49 ^ { 1 / 2 } \\[5pt]\)
    3. \(\left( \dfrac { 1 } { 4 } \right) ^ { 1 / 2 } \\[5pt]\)
    4. \(\left( \dfrac { 4 } { 9 } \right) ^ { 1 / 2 } \)
    1. \(4 ^ { - 1 / 2 } \\[2pt]\)
    2. \(9 ^ { - 1 / 2 } \\[2pt]\)
    3. \(\left( \dfrac { 1 } { 4 } \right) ^ { - 1 / 2 } \\[5pt]\)
    4. \(\left( \dfrac { 1 } { 16 } \right) ^ { - 1 / 2 } \)
    1. \(8 ^ { 1 / 3 } \\[2pt] \)
    2. \(125 ^ { 1 / 3 } \\[2pt]\)
    3. \(\left( \dfrac { 1 } { 27 } \right) ^ { 1 / 3 } \\[5pt]\)
    4. \(\left( \dfrac { 8 } { 125 } \right) ^ { 1 / 3 } \\[5pt]\)
    5. \(( - 27 ) ^ { 1 / 3 } \)
    1. \(( - 64 ) ^ { 1 / 3 } \\[5pt]\)
    2. \(16 ^ { 1 / 4 } \\[5pt]\)
    3. \(625 ^ { 1 / 4 } \\[5pt]\)
    4. \(81 ^ { - 1 / 4 } \\[5pt]\)
    5. \(16 ^ { - 1 / 4 } \)
    1. \(100,000 ^ { 1 / 5 } \\[5pt]\)
    2. \(( - 32 ) ^ { 1 / 5 } \\[5pt]\)
    3. \(\left( \dfrac { 1 } { 32 } \right) ^ { 1 / 5 } \\[5pt]\)
    4. \(\left( \dfrac { 1 } { 243 } \right) ^ { 1 / 5 } \)
    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.

    1. \(9 ^ { 3 / 2 } \\[5pt]\)
    2. \(4 ^ { 3 / 2 } \)
    1. \(8 ^ { 5 / 3 } \\[5pt]\)
    2. \(27 ^ { 2 / 3 } \)
    1. \(16 ^ { 3 / 2 } \\[5pt]\)
    2. \(32 ^ { 2 / 5 } \)
    1. \(\left( \dfrac { 1 } { 16 } \right) ^ { 3 / 4 } \\[5pt]\)
    2. \(\left( \dfrac { 1 } { 81 } \right) ^ { 3 / 4 } \)
    1. \(( - 27 ) ^ { 2 / 3 } \\[5pt]\)
    2. \( ( - 27 ) ^ { 4 / 3 } \)
    1. \(( - 32 ) ^ { 3 / 5 } \\[5pt]\)
    2. \(( - 32 ) ^ { 4 / 5 } \)
    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.

    1. \(5 ^ { 3 / 2 } \cdot 5 ^ { 1 / 2 } \\[5pt]\)
    2. \(3 ^ { 2 / 3 } \cdot 3 ^ { 7 / 3 } \\[5pt]\)
    3. \(5 ^ { 1 / 2 } \cdot 5 ^ { 1 / 3 } \\[5pt]\)
    4. \(2 ^ { 1 / 6 } \cdot 2 ^ { 3 / 4 } \)
    1. \(y ^ { 1 / 4 } \cdot y ^ { 2 / 5 } \\[5pt]\)
    2. \(x ^ { 1 / 2 } \cdot x ^ { 1 / 4 } \\[5pt]\)
    3. \((u^{12}v^{18})^{\tfrac{1}{6}} \\[5pt]\)
    4. ​​​​​​ \((r^{9}s^{12})^{\tfrac{1}{3}} \)
    1. \(\left( 8 ^ { 1 / 2 } \right) ^ { 2 / 3 } \\[5pt]\)
    2. \(\left( 3 ^ { 6 } \right) ^ { 2 / 3 } \\[5pt]\)
    3. \(\left( x ^ { 2 / 3 } \right) ^ { 1 / 2 } \\[5pt]\)
    4. \(\left( y ^ { 3 / 4 } \right) ^ { 4 / 5 } \)
    1. \(\left( y ^ { 8 } \right) ^ { - 1 / 2 } \\[5pt]\)
    2. \(\left( y ^ { 6 } \right) ^ { - 2 / 3 } \\[5pt]\)
    3. \(\left( 4 x ^ { 2 } y ^ { 4 } \right) ^ { 1 / 2 } \\[5pt]\)
    4. \(\left( 9 x ^ { 6 } y ^ { 2 } \right) ^ { 1 / 2 } \)
    1. \(\left( 2 x ^ { 1 / 3 } y ^ { 2 / 3 } \right) ^ { 3 } \\[5pt]\)
    2. \(\left( 8 x ^ { 3 / 2 } y ^ { 1 / 2 } \right) ^ { 2 } \\[5pt]\)
    3. \(\left( 36 x ^ { 4 } y ^ { 2 } \right) ^ { - 1 / 2 } \\[5pt]\)
    4. \(\left( 8 x ^ { 3 } y ^ { 6 } z ^ { - 3 } \right) ^ { - 1 / 3 } \)
    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.

    1. \(\left(27 q^{\tfrac{3}{2}}\right)^{\tfrac{4}{3}} \\[5pt]\)
    2. \(\left(64 s^{\tfrac{3}{7}}\right)^{\tfrac{1}{6}} \\[5pt]\)
    3. \(\left(a^{\tfrac{1}{3}} b^{\tfrac{2}{3}}\right)^{\tfrac{3}{2}} \)
    1. \( \left( m^{\tfrac{4}{3}} n^{\tfrac{1}{2}}\right)^{\tfrac{3}{4}} \\[5pt]\)
    2. \(\left(16 u^{\tfrac{1}{3}}\right)^{\tfrac{3}{4}} \\[5pt]\)
    3. \(\left(625 n^{\tfrac{8}{3}}\right)^{\tfrac{3}{4}} \)
    1. \(\left(4 p^{\tfrac{1}{3}} q^{\tfrac{1}{2}}\right)^{\tfrac{3}{2}} \\[5pt]\)
    2. \(\left(9 x^{\tfrac{2}{5}} y^{\tfrac{3}{5}}\right)^{\tfrac{5}{2}} \\[5pt]\)
    3. \(\left( 16 x ^ { 2 } y ^ { - 1 / 3 } z ^ { 2 / 3 } \right) ^ { - 3 / 2 } \)
    1. \(\left( 81 x ^ { 8 } y ^ { - 4 / 3 } z ^ { - 4 } \right) ^ { - 3 / 4 } \\[5pt]\)
    2. \(\left( 100 a ^ { - 2 / 3 } b ^ { 4 } c ^ { - 3 / 2 } \right) ^ { - 1 / 2 } \\[5pt]\)
    3. \(\left( 125 a ^ { 9 } b ^ { - 3 / 4 } c ^ { - 1 } \right) ^ { - 1 / 3 } \)
    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.

    1. \(\dfrac { 5 ^ { 11 / 3 } } { 5 ^ { 2 / 3 } } \\[5pt]\)
    2. \(\dfrac { 2 ^ { 9 / 2 } } { 2 ^ { 1 / 2 } } \\[5pt]\)
    3. \(\dfrac { 2 a ^ { 2 / 3 } } { a ^ { 1 / 6 } } \\[5pt]\)
    4. \(\dfrac { 3 b ^ { 1 / 2 } } { b ^ { 1 / 3 } }  \)
    1. \(\dfrac{r^{\tfrac{5}{2}} \cdot r^{-\tfrac{1}{2}}}{r^{-\tfrac{3}{2}}} \\[5pt]\)
    2. \(\dfrac{a^{\tfrac{3}{4}} \cdot a^{-\tfrac{1}{4}}}{a^{-\tfrac{10}{4}}} \\[5pt]\)
    3. \(\dfrac{c^{\tfrac{5}{3}} \cdot c^{-\tfrac{1}{3}}}{c^{-\tfrac{2}{3}}}  \)
    1. \(\dfrac{m^{\tfrac{7}{4}} \cdot m^{-\tfrac{5}{4}}}{m^{-\tfrac{2}{4}}} \\[16pt]\)
    2. 113 \(\dfrac { y ^ { 1 / 2 } y ^ { 2 / 3 } } { y ^ { 1 / 6 } } \\[16pt]\)
    3. \(\dfrac { x ^ { 2 / 5 } x ^ { 1 / 2 } } { x ^ { 1 / 10 } } \)
    1. \(\dfrac { x y } { x ^ { 1 / 2 } y ^ { 1 / 3 } }  \)
    2. \(\dfrac { x ^ { 5 / 4 } y } { x y ^ { 2 / 5 } }  \)
    3. \(\dfrac { 49 a ^ {5/7 } b ^ { 3 / 2 } } { 7 a ^ { 3 /7 } b ^ { 1 / 4 } } \\[5pt]\)
    4. \(\dfrac { 16 a ^ { 5 / 6 } b ^ { 5 / 4 } } { 8 a ^ { 1 / 2 } b ^ { 2 / 3 } } \)
    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.

    1. \(\left( \dfrac { a ^ { 3 / 4 } } { a ^ { 1 / 2 } } \right) ^ { 4 / 3 } \\[6pt]  \)
    2. \(\left( \dfrac { b ^ { 4 / 5 } } { b ^ { 1 / 10 } } \right) ^ { 10 / 3 } \\[6pt]\) 
    3. \(\left( \dfrac { 4 x ^ { 2 / 3 } } { y ^ { 4 } } \right) ^ { 1 / 2 } \\[6pt]  \)
    4. \(\left( \dfrac { 27 x ^ { 3 / 4 } } { y ^ { 9 } } \right) ^ { 1 / 3 }  \)
    1. \(\left( \dfrac { 27 x ^ { 3 / 4 } } { y ^ { 9 } } \right) ^ { 1 / 3 } \\[2pt] \)
    2. \(\left(\dfrac{36 s^{\tfrac{1}{5}} t^{-\tfrac{3}{2}}}{s^{-\tfrac{9}{5}} t^{\tfrac{1}{2}}}\right)^{\tfrac{1}{2}} \\[2pt]\)
    3. \(\left(\dfrac{27 b^{\tfrac{2}{3}} c^{-\tfrac{5}{2}}}{b^{-\tfrac{7}{3}} c^{\tfrac{1}{2}}}\right)^{\tfrac{1}{3}} \)
    1. \(\left(\dfrac{8 x^{\tfrac{5}{3}} y^{-\tfrac{1}{2}}}{27 x^{-\tfrac{4}{3}} y^{\tfrac{5}{2}}}\right)^{\tfrac{1}{3}} \\[1pt]\)
    2. \(\left(\dfrac{16 m^{\tfrac{1}{5}} n^{\tfrac{3}{2}}}{81 m^{\tfrac{9}{5}} n^{-\tfrac{1}{2}}}\right)^{\tfrac{1}{4}} \\[6pt] \)
    3. \(\dfrac { \left( 9 x ^ { 2 / 3 } y ^ { 6 } \right) ^ { 3 / 2 } } { x ^ { 1 / 2 } y }  \)
    1. \(\dfrac { \left( 125 x ^ { 3 } y ^ { 3 / 5 } \right) ^ { 2 / 3 } } { x y ^ { 1 / 3 } } \\[16pt]\)
    2. \(\dfrac { \left( 27 a ^ { 1 / 4 } b ^ { 3 / 2 } \right) ^ { 2 / 3 } } { a ^ { 1 / 6 } b ^ { 1 / 2 } } \\[16pt] \)
    3. \(\dfrac { \left( 25 a ^ { 2 / 3 } b ^ { 4 / 3 } \right) ^ { 3 / 2 } } { a ^ { 1 / 6 } b ^ { 1 / 3 } } \)
    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.

    1. \(\sqrt [ 3 ] { 9 } \cdot \sqrt [ 5 ] { 3 } \\[5pt]\)
    2. \(\sqrt { 5 } \cdot \sqrt [ 5 ] { 25 } \\[5pt]\)
    3. \(\sqrt { x } \cdot \sqrt [ 3 ] { x } \)
    1. \(\sqrt { y } \cdot \sqrt [ 4 ] { y } \\[5pt]\)
    2. \(\sqrt [ 3 ] { x ^ { 2 } } \cdot \sqrt [ 4 ] { x } \\[5pt]\)
    3. \(\sqrt [ 5 ] { x ^ { 3 } } \cdot \sqrt [ 3 ] { x } \)
    1. \(\dfrac { \sqrt [ 3 ] { 100 } } { \sqrt { 10 } } \\[5pt]\)
    2. \(\dfrac { \sqrt [ 5 ] { 16 } } { \sqrt [ 3 ] { 4 } } \)
    1. \(\dfrac { \sqrt [ 3 ] { a ^ { 2 } } } { \sqrt { a } } \\[5pt]\)
    2. \(\dfrac { \sqrt [ 5 ] { b ^ { 4 } } } { \sqrt [ 3 ] { b } } \)
    1. \(\dfrac { \sqrt [ 3 ] { x ^ { 2 } } } { \sqrt [ 5 ] { x ^ { 3 } } } \\[5pt]\)
    2. \(\dfrac { \sqrt [ 4 ] { x ^ { 3 } } } { \sqrt [ 3 ] { x ^ { 2 } } } \)
    1. \(\sqrt { \sqrt [ 5 ] { 16 } } \\[5pt]\)
    2. \(\sqrt { \sqrt [ 3 ] { 9 } } \\[5pt]\)
    3. \(\sqrt [ 3 ] { \sqrt [ 5 ] { 2 } } \)
    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 } )

    .\( \bigstar \)


    0.4e: Exercises - Rational Exponents is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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