Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite [latex]\sqrt{\frac{5}{2}}[/latex] as [latex]\frac{\sqrt{5}}{\sqrt{2}}[/latex].
The Quotient Rule for Simplifying Square Roots
The square root of the quotient [latex]\dfrac{a}{b}[/latex] is equal to the quotient of the square roots of [latex]a[/latex] and [latex]b[/latex], where [latex]b\ne 0[/latex].
How To: Given a radical expression, use the quotient rule to simplify it.
Write the radical expression as the quotient of two radical expressions.
Simplify the numerator and denominator.
Simplify the following radical expressions.
[latex]\sqrt{\dfrac{5}{36}}[/latex]
[latex]\sqrt{\dfrac{2{x}^{2}}{9{y}^{4}}}[/latex]
[latex]\begin{align*} \sqrt{\frac{5}{36}} &= \frac{\sqrt{5}}{\sqrt{36}} & \text{Separate the square root of the numerator and the denominator.} \\ &= \frac{\sqrt{5}}{6} & \text{Simplify the square root of the denominator.} \end{align*}[/latex]
[latex]\begin{align*} \sqrt{\frac{2x^2}{9y^4}} &= \frac{\sqrt{2x^2}}{\sqrt{9y^4}} & \text{Separate the square root of the numerator and the denominator.} \\ &= \frac{\sqrt{2} \cdot \sqrt{x^2}}{\sqrt{9} \cdot \sqrt{y^4}} & \text{Separate the square roots of the factors.} \\ &= \frac{\sqrt{2} \cdot |x|}{3 \cdot y^2} & \text{Simplify each square root, where } \sqrt{x^2} = |x| \text{ and } \sqrt{y^4} = |y|^2 = y^2. \\ &= \frac{|x|\sqrt{2}}{3y^2} & \text{Rearrange the terms for clarity.} \end{align*}[/latex]