1Cademy - Simplifying $$\sqrt[3]{8}$$, $$\sqrt[4]{81}$$, and $$\sqrt[5]{32}$$

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$n$ th Root of a Number

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Simplifying $\sqrt[3]{8}$ , $\sqrt[4]{81}$ , and $\sqrt[5]{32}$

Evaluate three higher roots of positive integers by recognizing each radicand as a perfect power of a small integer.

ⓐ $\sqrt[3]{8} = 2$ : Ask which number cubed equals $8$ . Since $2^3 = 8$ , the cube root of $8$ is $2$ .

ⓑ $\sqrt[4]{81} = 3$ : Ask which number raised to the fourth power equals $81$ . Since $3^4 = 81$ , the fourth root of $81$ is $3$ .

ⓒ $\sqrt[5]{32} = 2$ : Ask which number raised to the fifth power equals $32$ . Since $2^5 = 32$ , the fifth root of $32$ is $2$ .

In each case the strategy is the same: identify the base whose $n$ th power equals the radicand, then that base is the value of the radical. Familiarity with the powers of small integers — such as $2^3 = 8$ , $3^4 = 81$ , and $2^5 = 32$ — makes simplifying higher roots straightforward.

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Updated 2026-05-01

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