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.

$\sqrt{\frac{50x^5y^3}{72x^4y}}$

Step 1 — Simplify the fraction in the radicand. Divide out the common factors. The coefficients $50$ and $72$ share a factor of $2$, leaving $\frac{25}{36}$. Apply the Quotient Property for Exponents to the variables: $\frac{x^5}{x^4} = x$ and $\frac{y^3}{y} = y^2$. $\sqrt{\frac{25xy^2}{36}}$

Step 2 — Rewrite using the Quotient Property. Split the single radical into a quotient of two separate radicals: $\frac{\sqrt{25xy^2}}{\sqrt{36}}$

Step 3 — Simplify the radicals in the numerator and the denominator. The denominator is a perfect square: $\sqrt{36} = 6$. The numerator has perfect square factors $25$ and $y^2$, with a remaining factor of $x$: $\frac{\sqrt{25y^2} \cdot \sqrt{x}}{6}$

Step 4 — Simplify. Evaluate the perfect square roots: $\sqrt{25y^2} = 5/y/$. Multiply by the remaining radical: $\frac{5/y/\sqrt{x}}{6}$

The simplified result is $\frac{5/y/\sqrt{x}}{6}$. By simplifying the complex fraction inside the radical first, the denominator reduces to a perfect square, making the remaining steps straightforward.