0.04e: Exercises - Rational Exponents

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- 0.04: Review - Rational Exponents
- 0.05: Review - Factoring

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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:

$10 ^ { 1 / 2 }$

$3 ^ { 1 / 3 }$

$5 ^ { 2 / 3 }$

$7 ^ { 2 / 3 }$

$x ^ { 1 / 5 }$

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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A: Radical to Exponential Notation

B: Exponential to Radical Notation.

C: Exponential to Radical Form; then Simplify.

D: Exponential Operations. PRODUCTS and POWERS of Products

E: Exponential Operations. QUOTIENTS and POWERS of Quotients

F: Radical to Exponential Form Operations.

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