1Cademy - $$n$$th Root of a Number

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Square of a Number

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$n$ th Root of a Number

The concept of a square root generalizes to roots of any order. If $b^n = a$ , then $b$ is called an $n$ th root of $a$ . The principal $n$ th root of $a$ is written using radical notation as $\sqrt[n]{a}$ .

This definition extends the familiar square root — where $n = 2$ and the condition is $b^2 = a$ — to higher-order roots:

When $n = 3$ , $\sqrt[3]{a}$ is the cube root of $a$ .
When $n = 4$ , $\sqrt[4]{a}$ is the fourth root of $a$ .
When $n = 5$ , $\sqrt[5]{a}$ is the fifth root of $a$ .

For example, since $4^3 = 64$ , we have $\sqrt[3]{64} = 4$ ; since $3^4 = 81$ , we have $\sqrt[4]{81} = 3$ ; and since $(-2)^5 = -32$ , we have $\sqrt[5]{-32} = -2$ .

By convention, the index is not written for square roots — we write $\sqrt{a}$ rather than $\sqrt[2]{a}$ . Similarly, just as the term "cubed" describes $b^3$ , the term "cube root" describes $\sqrt[3]{a}$ .

Updated 2026-05-14

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