How to Evaluate Integers Raised to Rational Exponents

How to evaluate integers raised to rational exponents, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

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 \(\color{blue}{16^{\frac{1}{2}}}\).

Answers

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

Frequently Asked Questions

How do you evaluate integers raised to rational exponents?

Read the fraction in the exponent as root over power: the denominator tells you which root to take and the numerator tells you which power to raise it to. For 27^(2/3), take the cube root of 27 to get 3, then square it to get 9. Taking the root first keeps the numbers small, so you rarely need a calculator.

How do you evaluate rational exponents?

Rewrite the exponent as a radical: a^(m/n) = (ⁿ√a)^m. Then simplify the root and apply the power. For example, 16^(3/4) = (⁴√16)³ = 2³ = 8. If the exponent is negative, flip the base first: 16^(−3/4) = 1/16^(3/4) = 1/8. Simplifying the fraction in the exponent first, such as 4/6 to 2/3, also helps.

How do you evaluate integers raised to positive rational exponents?

With a positive rational exponent, the answer is simply a root followed by a power, with no reciprocal needed. Find the root named by the denominator, then raise it to the numerator. 25^(3/2) = (√25)³ = 5³ = 125, and 32^(3/5) = (⁵√32)³ = 2³ = 8. When the numerator is 1, as in 81^(1/4), the answer is just the root: 3.

How do you evaluate numbers raised to powers?

Multiply the base by itself as many times as the exponent says: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Watch the parentheses with negatives: (−3)⁴ = 81, but −3⁴ = −81 because only the 3 is raised to the power. Any nonzero number to the power 0 is 1. Rational exponents build on these rules by adding a root.

Can a calculator evaluate integers raised to rational exponents?

Yes, most scientific and graphing calculators can, but you must put the fraction in parentheses. Type 8^(2/3) to get 4; typing 8^2/3 instead gives 64 ÷ 3 ≈ 21.33, which is wrong. It is still worth knowing the root-then-power method, because many problems use perfect powers like 27, 32 or 64 that are faster to do by hand.

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