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Algebra II

Rational Exponents

Lesson

A rational exponentis just another way to write a radical. The denominator of the exponent says “take a root.”

x1/n=xnx^{1/n} = \sqrt[n]{x}

For example, x1/2=xx^{1/2} = \sqrt{x} (square root) and x1/3=x3x^{1/3} = \sqrt[3]{x} (cube root).

For a non-unit numerator, raise to that power too:

xm/n=xmn=(xn)mx^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m

So x2/3=x23=(x3)2x^{2/3} = \sqrt[3]{x^2} = (\sqrt[3]{x})^2.

Negative rational exponents reciprocate, just like with integers:

x−m/n=1xm/nx^{-m/n} = \frac{1}{x^{m/n}}

Worked example 1

Evaluate:

81/38^{1/3}

The cube root of 8. Since 23=82^3 = 8, the answer is 2.

Worked example 2

163/416^{3/4}

Take the 4th root first, then cube. The 4th root of 16 is 2 (since 2⁴ = 16), and 2³ = 8.

How to type your answer

For these problems, the answer is a number. Just type the numeric value (you can use fractions like 1/4). Examples: 2, 8, 1/4, 1/27.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

91/29^{1/2}

Problem 2

251/225^{1/2}

Problem 3

81/38^{1/3}

Problem 4

271/327^{1/3}

Practice

Standard problems matching the lesson.

Problem 5

161/216^{1/2}

Problem 6

641/264^{1/2}

Problem 7

641/364^{1/3}

Problem 8

161/416^{1/4}

Problem 9

811/481^{1/4}

Problem 10

321/532^{1/5}

Problem 11

82/38^{2/3}

Problem 12

163/416^{3/4}

Problem 13

272/327^{2/3}

Problem 14

93/29^{3/2}

Problem 15

43/24^{3/2}

Problem 16

322/532^{2/5}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

16−1/216^{-1/2}

Problem 18

8−1/38^{-1/3}

Problem 19

27−2/327^{-2/3}

Problem 20

16−3/416^{-3/4}

Problem 21

1252/3125^{2/3}

Problem 22

642/364^{2/3}

Practice

Standard problems matching the lesson.

Problem 23

Investment doubles every 7 years. After 14 years multiplier is 2^(14/7). Evaluate.

Problem 24

Algorithm runs in n^(2/3) time. For n = 27, evaluate 27^(2/3).

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

A scaling factor is 64^(2/3). Evaluate.

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Quiz

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