1Cademy - Simplifying $$(-64)^{\frac{1}{3}}$$, $$-64^{\frac{1}{3}}$$, and $$(64)^{-\frac{1}{3}}$$

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

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Simplifying $(-64)^{\frac{1}{3}}$ , $-64^{\frac{1}{3}}$ , and $(64)^{-\frac{1}{3}}$

Simplify three expressions that involve the number $64$ , a rational exponent $\frac{1}{3}$ , and a negative sign placed in three different positions — illustrating how sign placement changes the meaning and result.

ⓐ $(-64)^{\frac{1}{3}} = -4$ : The parentheses make $-64$ the base. Rewrite as a cube root: $(-64)^{\frac{1}{3}} = \sqrt[3]{-64}$ . Since $(-4)^3 = -64$ , the radicand is a perfect cube of $-4$ , so $\sqrt[3]{(-4)^3} = -4$ .

ⓑ $-64^{\frac{1}{3}} = -4$ : Without parentheses, the exponent $\frac{1}{3}$ applies only to $64$ , not to the negative sign. This means the expression equals $-(64^{\frac{1}{3}})$ . Rewrite $64^{\frac{1}{3}}$ as a cube root: $-\sqrt[3]{64}$ . Since $4^3 = 64$ , the cube root is $4$ , so $-\sqrt[3]{4^3} = -4$ .

ⓒ $(64)^{-\frac{1}{3}} = \frac{1}{4}$ : The exponent is negative, so apply the negative exponent property $a^{-n} = \frac{1}{a^n}$ first: $(64)^{-\frac{1}{3}} = \frac{1}{64^{\frac{1}{3}}}$ . Rewrite as $\frac{1}{\sqrt[3]{64}}$ . Since $4^3 = 64$ , the cube root is $4$ , giving $\frac{1}{\sqrt[3]{4^3}} = \frac{1}{4}$ .

Although parts ⓐ and ⓑ both yield $-4$ , they arrive there by different reasoning: in ⓐ the negative sign is part of the base being cubed, while in ⓑ the negation is applied after the cube root is evaluated. Part ⓒ produces a completely different result — the negative exponent creates a reciprocal, not a negative number.

Updated 2026-04-21

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OpenStax Elementary Algebra

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