1Cademy - Simplifying $$\sqrt[3]{-64}$$, $$\sqrt[4]{-16}$$, and $$\sqrt[5]{-243}$$

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Properties of $\sqrt[n]{a}$

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

Evaluate three higher roots of negative numbers, illustrating how the parity of the index determines whether the result is a real number.

ⓐ $\sqrt[3]{-64} = -4$ : The index $3$ is odd, so the cube root of a negative number is real. Since $(-4)^3 = -64$ , the cube root of $-64$ is $-4$ .

ⓑ $\sqrt[4]{-16}$ is not a real number: The index $4$ is even, and the radicand is negative. No real number raised to the fourth power can produce a negative result — even powers of any real number are always non-negative. Therefore $\sqrt[4]{-16}$ does not exist in the real numbers.

ⓒ $\sqrt[5]{-243} = -3$ : The index $5$ is odd, so the fifth root of a negative number is real. Since $(-3)^5 = -243$ , the fifth root of $-243$ is $-3$ .

Parts ⓐ and ⓒ both yield real (negative) results because odd-index roots of negative numbers exist. Part ⓑ is not real because even-index roots of negative numbers do not exist in the real number system. Checking whether the index is even or odd is always the first step when a negative radicand appears.

Updated 2026-05-01

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