Algebra II

Rational Exponents

Lesson

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

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

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

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

x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m

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

Negative rational exponents reciprocate, just like with integers:

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

Worked example 1

Evaluate:

8^{1/3}

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

Worked example 2

16^{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.

Work through these. Stuck? Click Get a hint.

Quick problems to get going.

Problem 1

9^{1/2}

Problem 2

25^{1/2}

Problem 3

8^{1/3}

Problem 4

27^{1/3}

Standard problems matching the lesson.

Problem 5

16^{1/2}

Problem 6

64^{1/2}

Problem 7

64^{1/3}

Problem 8

16^{1/4}

Problem 9

81^{1/4}

Problem 10

32^{1/5}

Problem 11

8^{2/3}

Problem 12

16^{3/4}

Problem 13

27^{2/3}

Problem 14

9^{3/2}

Problem 15

4^{3/2}

Problem 16

32^{2/5}

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

Problem 17

16^{-1/2}

Problem 18

8^{-1/3}

Problem 19

27^{-2/3}

Problem 20

16^{-3/4}

Problem 21

125^{2/3}

Problem 22

64^{2/3}