1Cademy - Simplifying $$25^{\frac{1}{2}}$$, $$64^{\frac{1}{3}}$$, and $$256^{\frac{1}{4}}$$

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Rational Exponent $a^{\frac{1}{n}}$

Example

Simplifying $25^{\frac{1}{2}}$ , $64^{\frac{1}{3}}$ , and $256^{\frac{1}{4}}$

Evaluate three numerical expressions with rational exponents of the form $\frac{1}{n}$ by converting each to radical form and simplifying.

ⓐ $25^{\frac{1}{2}} = 5$ : Rewrite the rational exponent as a square root: $25^{\frac{1}{2}} = \sqrt{25}$ . Since $5^2 = 25$ , the result is $\sqrt{25} = 5$ .

ⓑ $64^{\frac{1}{3}} = 4$ : Rewrite as a cube root: $64^{\frac{1}{3}} = \sqrt[3]{64}$ . Recognize that $64$ is a perfect cube because $4^3 = 64$ . Express the radicand accordingly: $\sqrt[3]{4^3} = 4$ .

ⓒ $256^{\frac{1}{4}} = 4$ : Rewrite as a fourth root: $256^{\frac{1}{4}} = \sqrt[4]{256}$ . Recognize that $256$ is a perfect fourth power because $4^4 = 256$ . Therefore $\sqrt[4]{4^4} = 4$ .

In each case the procedure is the same: use the rule $a^{\frac{1}{n}} = \sqrt[n]{a}$ to convert the expression to radical form, then identify the base whose $n$ th power equals the radicand. Recognizing perfect powers — perfect squares, perfect cubes, and perfect fourth powers — is the key skill that makes these simplifications quick.

Updated 2026-05-01

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