1Cademy - Simplifying $$27^{\frac{2}{3}}$$ and $$4^{\frac{3}{2}}$$

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

Example

Simplifying $27^{\frac{2}{3}}$ and $4^{\frac{3}{2}}$

Evaluate numerical expressions with rational exponents of the form $\frac{m}{n}$ by converting them to radical form using $a^{\frac{m}{n}} = \left(\sqrt[n]{a} ight)^m$ . Taking the root before raising to the power keeps the intermediate numbers small.

ⓐ $27^{\frac{2}{3}} = 9$ : The denominator of the exponent is $3$ , which gives a cube root, and the numerator $2$ is the power. Rewrite the expression: $27^{\frac{2}{3}} = \left(\sqrt[3]{27} ight)^2$ . Since $3^3 = 27$ , the cube root of $27$ is $3$ . Evaluate $3^2 = 9$ .

ⓑ $4^{\frac{3}{2}} = 8$ : The denominator of the exponent is $2$ , which gives a square root, and the numerator $3$ is the power. Rewrite the expression: $4^{\frac{3}{2}} = \left(\sqrt{4} ight)^3$ . Since $2^2 = 4$ , the square root of $4$ is $2$ . Evaluate $2^3 = 8$ .

Updated 2026-05-01

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OpenStax Intermediate Algebra

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