1Cademy - Square Roots of Variable Expressions

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Square Roots of Variable Expressions

When the radicand of a square root contains a variable raised to an even power, the expression can be simplified by applying the Power Property of Exponents in reverse. Because $(a^m)^2 = a^{2m}$ , it follows that:

$\sqrt{a^{2m}} = a^m$

In words: to take the square root of a variable raised to an even exponent, divide the exponent by 2. The same idea extends to expressions that include a numerical perfect-square coefficient multiplied by a variable power — take the square root of the coefficient and halve the variable's exponent separately. For example:

$\sqrt{9x^2} = 3x$ because $(3x)^2 = 9x^2$
$\sqrt{25u^8} = 5u^4$ because $(5u^4)^2 = 25u^8$
$\sqrt{16r^{20}} = 4r^{10}$ because $(4r^{10})^2 = 16r^{20}$
$\sqrt{196q^{36}} = 14q^{18}$ because $(14q^{18})^2 = 196q^{36}$

To ensure the radical sign yields the principal (non-negative) square root, the variable must represent a non-negative number ( $x \geq 0$ ). In practice, this assumption is often stated once at the beginning of the work and then left implicit throughout the remaining problems.

Updated 2026-04-21

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