1Cademy - Simplifying $$\sqrt[3]{-27}$$, $$\sqrt[4]{-81}$$, $$\sqrt[3]{-625}$$, and $$\sqrt[4]{-324}$$

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Simplifying $\sqrt[3]{-64}$ , $\sqrt[4]{-16}$ , and $\sqrt[5]{-243}$

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Simplifying $\sqrt[3]{-27}$ , $\sqrt[4]{-81}$ , $\sqrt[3]{-625}$ , and $\sqrt[4]{-324}$

Evaluate higher roots of negative numbers by observing whether the index is odd or even. ⓐ $\sqrt[3]{-27} = -3$ : The index $3$ is odd, so the cube root of a negative number is a real number. Since $(-3)^3 = -27$ , the cube root is $-3$ . ⓑ $\sqrt[4]{-81}$ is not a real number: The index $4$ is even, and there is no real number $n$ where $n^4 = -81$ . Even-index roots of negative numbers do not exist in the real number system. ⓒ $\sqrt[3]{-625} = -5\sqrt[3]{5}$ : The index $3$ is odd. Rewrite the radicand using its largest perfect cube factor: $-625 = -125 \cdot 5$ . Since $(-5)^3 = -125$ , the expression simplifies to $-5\sqrt[3]{5}$ . ⓓ $\sqrt[4]{-324}$ is not a real number: Because the index $4$ is even and the radicand is negative, the root is not a real number.

Updated 2026-06-05

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OpenStax Intermediate Algebra

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