To simplify a square root whose radicand is a fraction containing numerical coefficients and variables, first reduce the fraction under the radical, then apply the Quotient Property of Radical Expressions.

$\sqrt{\frac{18p^5q^7}{32pq^2}}$

Step 1 — Simplify the fraction in the radicand. Divide out the common factors. The coefficients $18$ and $32$ share a factor of $2$, leaving $\frac{9}{16}$. Apply the Quotient Property for Exponents to the variables: $\frac{p^5}{p} = p^4$ and $\frac{q^7}{q^2} = q^5$. $\sqrt{\frac{9p^4q^5}{16}}$

Step 2 — Rewrite using the Quotient Property. Split the single radical into a quotient of two separate radicals: $\frac{\sqrt{9p^4q^5}}{\sqrt{16}}$

Step 3 — Simplify the radicals in the numerator and the denominator. The denominator is a perfect square: $\sqrt{16} = 4$. The numerator has perfect square factors $9$, $p^4$, and $q^4$, with a remaining factor of $q$: $\frac{\sqrt{9p^4q^4} \cdot \sqrt{q}}{4}$

Step 4 — Simplify. Evaluate the perfect square roots: $\sqrt{9p^4q^4} = 3p^2q^2$. Multiply by the remaining radical: $\frac{3p^2q^2\sqrt{q}}{4}$

The simplified result is $\frac{3p^2q^2\sqrt{q}}{4}$. By simplifying the complex fraction inside the radical first, the denominator reduces to a perfect square, making the remaining steps straightforward.