How to Evaluate Integers Raised to Rational Exponents

Rational exponents combine roots and powers into a single compact notation. Once you learn the connection between rational exponents and radicals, every problem reduces to a straightforward root-and-power calculation. This lesson walks you through the rules step by step, complete with worked examples, two video walkthroughs, and practice exercises to prepare you for GED Math.

What Are Rational Exponents?

A rational exponent is a fraction used as an exponent. The general rule is:

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a^{\(\color{blue}{\frac{m}{n}}\)} = ( ⁿ√a )^m = ⁿ√(a^m)

• The denominator of the fraction tells you which root to take.
• The numerator of the fraction tells you the power to raise to.

For example, \(\color{blue}{8^{\frac{2}{3}}}\) means “take the cube root of 8, then square it”: \(\color{blue}{(^{3}\sqrt{8})^{2} = 2^{2} = 4}\).

Rules and Method for Rational Exponents

Rule 1: a^{\(\color{blue}{\frac{1}{n}}\)} = ⁿ√a (fractional exponent with numerator 1)

When the numerator is 1, the exponent is simply the nth root of the base.

• \(\color{blue}{27^{\frac{1}{3}} = ^{3}\sqrt{27} = 3}\)
• \(\color{blue}{16^{\frac{1}{4}} = \sqrt ^{4}16 = 2}\)
• \(\color{blue}{9^{\frac{1}{2}} = \sqrt{9} = 3}\)

Rule 2: a^{\(\color{blue}{\frac{m}{n}}\)} = (ⁿ√a)^m (take root first, then raise to power)

It is usually easier to take the root first and then apply the power, keeping numbers small.

• \(\color{blue}{8^{\frac{2}{3}} = (^{3}\sqrt{8})^{2} = 2^{2} = 4}\)
• \(\color{blue}{16^{\frac{3}{4}} = (\sqrt ^{4}16)^{3} = 2^{3} = 8}\)
• \(\color{blue}{32^{\frac{2}{5}} = (\sqrt ^{5}32)^{2} = 2^{2} = 4}\)

Rule 3: Negative rational exponents

A negative exponent means take the reciprocal: \(\color{blue}{a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}}}\).

• \(\color{blue}{8^{-\frac{1}{3}} = 1 / ^{3}\sqrt{8} = \frac{1}{2}}\)
• \(\color{blue}{4^{-\frac{3}{2}} = \frac{1}{(\sqrt{4})^{3}} = \frac{1}{8}}\)

Step-by-Step Summary

1. Rewrite the rational exponent as a radical: denominator → root index, numerator → power.
2. Evaluate the root first (keep the base a perfect power if possible).
3. Raise the result to the numerator power.
4. If the exponent is negative, take the reciprocal of the positive-exponent result.

Watch: How to Evaluate a Number Raised to a Rational Exponent

This focused lesson walks through the core method with concrete examples:

Worked Examples

Example 1: Evaluate \(\color{blue}{27^{\frac{2}{3}}}\).

Step 1: Denominator is 3, so take the cube root: \(\color{blue}{^{3}\sqrt{27} = 3}\).
Step 2: Numerator is 2, so square the result: \(\color{blue}{3^{2} = 9}\).
Answer: 9

Example 2: Evaluate \(\color{blue}{64^{\frac{2}{3}}}\).

Step 1: \(\color{blue}{^{3}\sqrt{64} = 4}\).
Step 2: \(\color{blue}{4^{2} = 16}\).
Answer: 16

Example 3: Evaluate \(\color{blue}{25^{\frac{3}{2}}}\).

Step 1: \(\color{blue}{\sqrt{25} = 5}\).
Step 2: \(\color{blue}{5^{3} = 125}\).
Answer: 125

Example 4: Evaluate \(\color{blue}{16^{\frac{3}{4}}}\).

Step 1: \(\color{blue}{\sqrt ^{4}16 = 2}\).
Step 2: \(\color{blue}{2^{3} = 8}\).
Answer: 8

More Practice: Rational Exponents Properties (Video)

This Khan Academy lesson covers the properties of rational exponents with additional examples:

Exercises

1. Evaluate \(\color{blue}{8^{\frac{1}{3}}}\).
2. Evaluate \(\color{blue}{4^{\frac{3}{2}}}\).
3. Evaluate \(\color{blue}{32^{\frac{2}{5}}}\).
4. Evaluate \(\color{blue}{81^{\frac{3}{4}}}\).
5. Evaluate \(\color{blue}{125^{\frac{2}{3}}}\).
6. Evaluate \(\color{blue}{16^{\frac{1}{2}}}\).

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Answers

1. \(\color{blue}{2}\)
2. \(\color{blue}{8}\)
3. \(\color{blue}{4}\)
4. \(\color{blue}{27}\)
5. \(\color{blue}{25}\)
6. \(\color{blue}{4}\)

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Frequently Asked Questions

What does a rational exponent mean?

A rational exponent is a fraction exponent. The denominator indicates which root to take and the numerator indicates the power. For example, \(\color{blue}{a^{\frac{m}{n}} = (^{n}\sqrt{a})^{m}}\).

Is it easier to take the root or the power first?

It is almost always easier to take the root first. That keeps the number smaller and makes the power calculation simpler. For \(\color{blue}{8^{\frac{2}{3}}}\), taking the cube root first gives 2, and then squaring gives 4 — far simpler than computing 8²=64 and then cubing root of 64.

Can rational exponents apply to negative bases?

Yes, but only when the root index is odd. For example, \(\color{blue}{(-8)^{\frac{1}{3}} = -2}\) because the cube root \(\color{blue}{\text{ of } -8}\) \(\color{blue}{\text{ is } -2}\). Even roots of negative numbers are not real numbers.

Related Topics

• Prime Factorization with Exponents
• Using Exponents to Write Powers of Ten
• Use Exponents to Write Multiplication Expressions
• Identifying Expressions and Equations
• Use Properties to Write Equivalent Expressions