0.04e: Exercises - Rational Exponents - Mathematics LibreTexts

A: Radical to Exponential Notation

Exercise $0.4.1$

$★$ Express using rational exponents.

$\sqrt{10}$
$\sqrt{6}$

$\sqrt[3]{3}$
$\sqrt[4]{5}$

$\sqrt[3]{5^{2}}$
$\sqrt[4]{2^{3}}$

$\sqrt[3]{49}$
$\sqrt[3]{9}$

$\sqrt[5]{x}$
$\sqrt[6]{x}$

$\sqrt[6]{x^{7}}$
$\sqrt[5]{x^{4}}$

$\frac{1}{\sqrt{x}}$
$\frac{1}{\sqrt[3]{x^{2}}}$

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 $0.4.2$

$★$ Express in radical form.

$10^{1 / 2}$
$11^{1 / 3}$

$7^{2 / 3}$
$2^{3 / 5}$

$x^{3 / 4}$
$x^{5 / 6}$

$x^{- 1 / 2}$
$x^{- 3 / 4}$

${(\frac{1}{x})}^{- 1 / 3}$
${(\frac{1}{x})}^{- 3 / 5}$

${(2 x + 1)}^{2 / 3}$
${(5 x - 1)}^{1 / 2}$

Answers to odd exercises:

15.

\sqrt{10}

17.

\sqrt[3]{49}

19.

\sqrt[4]{x^{3}}

21.

\frac{1}{\sqrt{x}}

23.

\sqrt[3]{x}

25.

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

C: Exponential to Radical Form; then Simplify.

Exercise $0.4.3$

$★$ Write as a radical and then simplify.

$64^{1 / 2}$
$49^{1 / 2}$
${(\frac{1}{4})}^{1 / 2}$
${(\frac{4}{9})}^{1 / 2}$

$4^{- 1 / 2}$
$9^{- 1 / 2}$
${(\frac{1}{4})}^{- 1 / 2}$
${(\frac{1}{16})}^{- 1 / 2}$

$8^{1 / 3}$
$125^{1 / 3}$
${(\frac{1}{27})}^{1 / 3}$
${(\frac{8}{125})}^{1 / 3}$
${(- 27)}^{1 / 3}$

${(- 64)}^{1 / 3}$
$16^{1 / 4}$
$625^{1 / 4}$
$81^{- 1 / 4}$
$16^{- 1 / 4}$

$100, 000^{1 / 5}$
${(- 32)}^{1 / 5}$
${(\frac{1}{32})}^{1 / 5}$
${(\frac{1}{243})}^{1 / 5}$

Answers to odd exercises:

27.

8

29.

\frac{1}{2}

31.

\frac{1}{2}

33.

2

35.

2

37.

\frac{1}{3}

39.

- 3

41.

2

43.

\frac{1}{3}

45.

10

47.

\frac{1}{2}

$★$ Write as a radical and then simplify.

$9^{3 / 2}$
$4^{3 / 2}$

$8^{5 / 3}$
$27^{2 / 3}$

$16^{3 / 2}$
$32^{2 / 5}$

${(\frac{1}{16})}^{3 / 4}$
${(\frac{1}{81})}^{3 / 4}$

${(- 27)}^{2 / 3}$
${(- 27)}^{4 / 3}$

${(- 32)}^{3 / 5}$
${(- 32)}^{4 / 5}$

Answers to odd exercises:

49.

27

51.

32

53.

64

55.

\frac{1}{8}

57.

9

59.

- 8

D: Exponential Operations. PRODUCTS and POWERS of Products

Exercise $0.4.4$

$★$ Perform the operations and simplify. Leave answers in exponential form.

$5^{3 / 2} \cdot 5^{1 / 2}$
$3^{2 / 3} \cdot 3^{7 / 3}$
$5^{1 / 2} \cdot 5^{1 / 3}$
$2^{1 / 6} \cdot 2^{3 / 4}$

$y^{1 / 4} \cdot y^{2 / 5}$
$x^{1 / 2} \cdot x^{1 / 4}$
${(u^{12} v^{18})}^{\frac{1}{6}}$
${(r^{9} s^{12})}^{\frac{1}{3}}$

${(8^{1 / 2})}^{2 / 3}$
${(3^{6})}^{2 / 3}$
${(x^{2 / 3})}^{1 / 2}$
${(y^{3 / 4})}^{4 / 5}$

${(y^{8})}^{- 1 / 2}$
${(y^{6})}^{- 2 / 3}$
${(4 x^{2} y^{4})}^{1 / 2}$
${(9 x^{6} y^{2})}^{1 / 2}$

${(2 x^{1 / 3} y^{2 / 3})}^{3}$
${(8 x^{3 / 2} y^{1 / 2})}^{2}$
${(36 x^{4} y^{2})}^{- 1 / 2}$
${(8 x^{3} y^{6} z^{- 3})}^{- 1 / 3}$

Answers to odd exercises:

61.

25

63.

5^{5 / 6}

65.

y^{13 / 20}

67.

u^{2} v^{3}

69.

2

71.

x^{1 / 3}

73.

\frac{1}{y^{4}}

75.

2 x y^{2}

77.

8 x y^{2}

79.

\frac{1}{6 x^{2} y}

$★$ Perform the operations and simplify. Leave answers in exponential form.

${(27 q^{\frac{3}{2}})}^{\frac{4}{3}}$
${(64 s^{\frac{3}{7}})}^{\frac{1}{6}}$
${(a^{\frac{1}{3}} b^{\frac{2}{3}})}^{\frac{3}{2}}$

${(m^{\frac{4}{3}} n^{\frac{1}{2}})}^{\frac{3}{4}}$
${(16 u^{\frac{1}{3}})}^{\frac{3}{4}}$
${(625 n^{\frac{8}{3}})}^{\frac{3}{4}}$

${(4 p^{\frac{1}{3}} q^{\frac{1}{2}})}^{\frac{3}{2}}$
${(9 x^{\frac{2}{5}} y^{\frac{3}{5}})}^{\frac{5}{2}}$
${(16 x^{2} y^{- 1 / 3} z^{2 / 3})}^{- 3 / 2}$

${(81 x^{8} y^{- 4 / 3} z^{- 4})}^{- 3 / 4}$
${(100 a^{- 2 / 3} b^{4} c^{- 3 / 2})}^{- 1 / 2}$
${(125 a^{9} b^{- 3 / 4} c^{- 1})}^{- 1 / 3}$

Answers to odd exercises:

81.

81 q^{2}

83.

a^{\frac{1}{2}} b

85.

8 u^{\frac{1}{4}}

87.

8 p^{\frac{1}{2}} q^{\frac{3}{4}}

85.

8 u^{\frac{1}{4}}

87.

8 p^{\frac{1}{2}} q^{\frac{3}{4}}

89.

\frac{y^{1 / 2}}{64 x^{3} z}

91.

\frac{a^{1 / 3} c^{3 / 4}}{10 b^{2}}

E: Exponential Operations. QUOTIENTS and POWERS of Quotients

Exercise $0.4.5$

$★$ Perform the operations and simplify. Leave answers in exponential form.

$\frac{5^{11 / 3}}{5^{2 / 3}}$
$\frac{2^{9 / 2}}{2^{1 / 2}}$
$\frac{2 a^{2 / 3}}{a^{1 / 6}}$
$\frac{3 b^{1 / 2}}{b^{1 / 3}}$

$\frac{r^{\frac{5}{2}} \cdot r^{- \frac{1}{2}}}{r^{- \frac{3}{2}}}$
$\frac{a^{\frac{3}{4}} \cdot a^{- \frac{1}{4}}}{a^{- \frac{10}{4}}}$
$\frac{c^{\frac{5}{3}} \cdot c^{- \frac{1}{3}}}{c^{- \frac{2}{3}}}$

$\frac{m^{\frac{7}{4}} \cdot m^{- \frac{5}{4}}}{m^{- \frac{2}{4}}}$
113 $\frac{y^{1 / 2} y^{2 / 3}}{y^{1 / 6}}$
$\frac{x^{2 / 5} x^{1 / 2}}{x^{1 / 10}}$

$\frac{x y}{x^{1 / 2} y^{1 / 3}}$
$\frac{x^{5 / 4} y}{x y^{2 / 5}}$
$\frac{49 a^{5 / 7} b^{3 / 2}}{7 a^{3 / 7} b^{1 / 4}}$
$\frac{16 a^{5 / 6} b^{5 / 4}}{8 a^{1 / 2} b^{2 / 3}}$

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}

$★$ Perform the operations and simplify. Leave answers in exponential form.

${(\frac{a^{3 / 4}}{a^{1 / 2}})}^{4 / 3}$
${(\frac{b^{4 / 5}}{b^{1 / 10}})}^{10 / 3}$
${(\frac{4 x^{2 / 3}}{y^{4}})}^{1 / 2}$
${(\frac{27 x^{3 / 4}}{y^{9}})}^{1 / 3}$

${(\frac{27 x^{3 / 4}}{y^{9}})}^{1 / 3}$
${(\frac{36 s^{\frac{1}{5}} t^{- \frac{3}{2}}}{s^{- \frac{9}{5}} t^{\frac{1}{2}}})}^{\frac{1}{2}}$
${(\frac{27 b^{\frac{2}{3}} c^{- \frac{5}{2}}}{b^{- \frac{7}{3}} c^{\frac{1}{2}}})}^{\frac{1}{3}}$

${(\frac{8 x^{\frac{5}{3}} y^{- \frac{1}{2}}}{27 x^{- \frac{4}{3}} y^{\frac{5}{2}}})}^{\frac{1}{3}}$
${(\frac{16 m^{\frac{1}{5}} n^{\frac{3}{2}}}{81 m^{\frac{9}{5}} n^{- \frac{1}{2}}})}^{\frac{1}{4}}$
$\frac{{(9 x^{2 / 3} y^{6})}^{3 / 2}}{x^{1 / 2} y}$

$\frac{{(125 x^{3} y^{3 / 5})}^{2 / 3}}{x y^{1 / 3}}$
$\frac{{(27 a^{1 / 4} b^{3 / 2})}^{2 / 3}}{a^{1 / 6} b^{1 / 2}}$
$\frac{{(25 a^{2 / 3} b^{4 / 3})}^{3 / 2}}{a^{1 / 6} b^{1 / 3}}$

Answers to odd exercises:

115.

a^{1 / 3}

117.

\frac{2 x^{1 / 3}}{y^{2}}

119.

\frac{6 s}{t}

121.

\frac{2 x}{3 y}

123.

27 x^{1 / 2} y^{8}

125.

9 b^{1 / 2}

F: Radical to Exponential Form Operations.

Exercise $0.4.6$

$★$ Rewrite in exponential form and then perform the operations and write the final answer in radical form.

$\sqrt[3]{9} \cdot \sqrt[5]{3}$
$\sqrt{5} \cdot \sqrt[5]{25}$
$\sqrt{x} \cdot \sqrt[3]{x}$

$\sqrt{y} \cdot \sqrt[4]{y}$
$\sqrt[3]{x^{2}} \cdot \sqrt[4]{x}$
$\sqrt[5]{x^{3}} \cdot \sqrt[3]{x}$

$\frac{\sqrt[3]{100}}{\sqrt{10}}$
$\frac{\sqrt[5]{16}}{\sqrt[3]{4}}$

$\frac{\sqrt[3]{a^{2}}}{\sqrt{a}}$
$\frac{\sqrt[5]{b^{4}}}{\sqrt[3]{b}}$

$\frac{\sqrt[3]{x^{2}}}{\sqrt[5]{x^{3}}}$
$\frac{\sqrt[4]{x^{3}}}{\sqrt[3]{x^{2}}}$

$\sqrt{\sqrt[5]{16}}$
$\sqrt{\sqrt[3]{9}}$
$\sqrt[3]{\sqrt[5]{2}}$

Answers to odd exercises:

131.

\sqrt[15]{3^{13}}

133.

\sqrt[6]{x^{5}}

135.

\sqrt[12]{x^{11}}

137.

\sqrt[6]{10}

139.

\sqrt[6]{a}

141.

\sqrt[15]{x}

143.

\sqrt[5]{4}

145.

\sqrt[15]{2}

8/16/2025

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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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