1Cademy - Simplifying $$9^{\frac{3}{2}}$$, $$125^{\frac{2}{3}}$$, and $$81^{\frac{3}{4}}$$

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

Example

Simplifying $9^{\frac{3}{2}}$ , $125^{\frac{2}{3}}$ , and $81^{\frac{3}{4}}$

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

ⓐ $9^{\frac{3}{2}} = 27$ : The denominator of the exponent is $2$ , so the radical is a square root. The numerator $3$ is the power. Rewrite: $9^{\frac{3}{2}} = \left(\sqrt{9}\right)^3$ . Since $\sqrt{9} = 3$ , evaluate $3^3 = 27$ .

ⓑ $125^{\frac{2}{3}} = 25$ : The denominator $3$ gives a cube root, and the numerator $2$ is the power. Rewrite: $125^{\frac{2}{3}} = \left(\sqrt[3]{125}\right)^2$ . Since $5^3 = 125$ , the cube root is $5$ . Evaluate $5^2 = 25$ .

ⓒ $81^{\frac{3}{4}} = 27$ : The denominator $4$ gives a fourth root, and the numerator $3$ is the power. Rewrite: $81^{\frac{3}{4}} = \left(\sqrt[4]{81}\right)^3$ . Since $3^4 = 81$ , the fourth root is $3$ . Evaluate $3^3 = 27$ .

In each case, the denominator of the exponent determines which root to take (square, cube, or fourth), and the numerator determines the power to apply afterward. Recognizing perfect powers — $9 = 3^2$ , $125 = 5^3$ , $81 = 3^4$ — is essential for evaluating the root step quickly.

Updated 2026-05-01

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