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

$\sqrt{\frac{48m^7n^2}{100m^5n^8}}$

Step 1 — Simplify the fraction in the radicand. Divide out common numerical factors. The coefficients $48$ and $100$ share a common factor of $4$, leaving $\frac{12}{25}$. Apply the Quotient Property for Exponents to the variables: $\frac{m^7}{m^5} = m^2$ in the numerator, and $\frac{n^2}{n^8} = \frac{1}{n^6}$ in the denominator. The expression becomes: $\sqrt{\frac{12m^2}{25n^6}}$

Step 2 — Rewrite using the Quotient Property. Split the single radical into a quotient of two separate radicals: $\frac{\sqrt{12m^2}}{\sqrt{25n^6}}$

Step 3 — Simplify the radicals in the numerator and the denominator. The denominator is a perfect square: $\sqrt{25n^6} = 5|n^3|$. For the numerator, factor $12m^2$ into a perfect square and a remaining factor: $\sqrt{4m^2 \cdot 3}$. Separate the radicals: $\sqrt{4m^2} \cdot \sqrt{3}$. Evaluate the perfect square: $2|m|\sqrt{3}$.

Step 4 — Simplify the final expression. $\frac{2|m|\sqrt{3}}{5|n^3|}$

The simplified result is $\frac{2|m|\sqrt{3}}{5|n^3|}$. Reducing the fraction under the radical first simplifies the variables and reveals a perfect square in the denominator.