1Cademy - Properties of $$\sqrt[n]{a}$$

Learn Before

$n$ th Root of a Number

Concept

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

Whether the $n$ th root of a number is a real number depends on two factors: the sign of the radicand $a$ and whether the index $n$ is even or odd.

When $n$ is even:

If $a \geq 0$ , then $\sqrt[n]{a}$ is a real number.
If $a < 0$ , then $\sqrt[n]{a}$ is not a real number.

This is because raising any real number to an even power always produces a non-negative result — no real number raised to an even power can be negative. For example, $\sqrt[4]{-16}$ is not a real number because no real number satisfies $(?)^4 = -16$ .

When $n$ is odd, $\sqrt[n]{a}$ is a real number for all values of $a$ — positive, negative, or zero. Odd powers preserve the sign of the base, so a negative number raised to an odd power remains negative. For example, $\sqrt[5]{-243} = -3$ because $(-3)^5 = -243$ .

This generalizes the familiar fact that the square root of a negative number is not real: square roots are the special case where the index $n = 2$ is even.

Additionally, for any integer $n \geq 2$ , the $n$ th root and the $n$ th power are inverse operations, but the result depends on whether the index is even or odd:

When $n$ is odd: $\sqrt[n]{a^n} = a$
When $n$ is even: $\sqrt[n]{a^n} = |a|$

The absolute value is needed in the even case because raising $a$ to an even power always produces a non-negative result (information about the sign of $a$ is lost), so the principal $n$ th root returns only the non-negative value $|a|$ . In the odd case, the sign is preserved throughout the process, so the result is simply $a$ without absolute value.

Updated 2026-05-01

Contributors are:

Who are from:

References

Learn Before

Related

Learn After