1Cademy - Simplifying $$\sqrt{x^6}$$ and $$\sqrt{y^{16}}$$

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Simplifying $\sqrt{x^6}$ and $\sqrt{y^{16}}$

Simplify two square roots whose radicands are variables raised to even powers, by dividing each exponent by 2.

ⓐ $\sqrt{x^6} = x^3$ : Since $(x^3)^2 = x^6$ , the expression $x^3$ is the value whose square equals $x^6$ . Therefore $\sqrt{x^6} = x^3$ .

ⓑ $\sqrt{y^{16}} = y^8$ : Since $(y^8)^2 = y^{16}$ , the expression $y^8$ is the value whose square equals $y^{16}$ . Therefore $\sqrt{y^{16}} = y^8$ .

In both cases the procedure is the same: recognize that the exponent under the radical is even, then halve it to obtain the simplified result. This works because the Power Property guarantees $(a^m)^2 = a^{2m}$ , so taking the square root reverses the doubling of the exponent.

Updated 2026-04-21

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OpenStax Elementary Algebra

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