1Cademy - Simplifying $$16^{-\frac{3}{2}}$$, $$32^{-\frac{2}{5}}$$, and $$4^{-\frac{5}{2}}$$

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

Example

Simplifying $16^{-\frac{3}{2}}$ , $32^{-\frac{2}{5}}$ , and $4^{-\frac{5}{2}}$

Simplify three numerical expressions whose exponents are negative rational numbers of the form $-\frac{m}{n}$ by combining the negative exponent rule $b^{-p} = \frac{1}{b^p}$ with conversion to radical form.

ⓐ $16^{-\frac{3}{2}} = \frac{1}{64}$ : First apply the negative exponent rule to make the exponent positive: $16^{-\frac{3}{2}} = \frac{1}{16^{\frac{3}{2}}}$ . Convert to radical form — the denominator $2$ gives a square root, and the numerator $3$ is the power: $\frac{1}{(\sqrt{16})^3}$ . Since $\sqrt{16} = 4$ , evaluate: $\frac{1}{4^3} = \frac{1}{64}$ .

ⓑ $32^{-\frac{2}{5}} = \frac{1}{4}$ : Apply the negative exponent rule: $32^{-\frac{2}{5}} = \frac{1}{32^{\frac{2}{5}}}$ . Convert to radical form — the denominator $5$ gives a fifth root, and the numerator $2$ is the power: $\frac{1}{(\sqrt[5]{32})^2}$ . Rewrite $32$ as $2^5$ so that $\sqrt[5]{2^5} = 2$ . Evaluate: $\frac{1}{2^2} = \frac{1}{4}$ .

ⓒ $4^{-\frac{5}{2}} = \frac{1}{32}$ : Apply the negative exponent rule: $4^{-\frac{5}{2}} = \frac{1}{4^{\frac{5}{2}}}$ . Convert to radical form — the denominator $2$ gives a square root, and the numerator $5$ is the power: $\frac{1}{(\sqrt{4})^5}$ . Since $\sqrt{4} = 2$ , evaluate: $\frac{1}{2^5} = \frac{1}{32}$ .

In every case, the two-step strategy is the same: first use the negative exponent rule to rewrite the expression as $\frac{1}{b^{\frac{m}{n}}}$ , then convert the positive rational exponent to radical form and simplify. The denominator of the fractional exponent determines which root to take, and the numerator determines the power to apply afterward.

Updated 2026-05-01

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