Simplify higher-order roots whose radicands are complex fractions by first reducing the fraction inside the radical, then applying the Quotient Property of $n$th Roots.

ⓑ $\sqrt[3]{\frac{16x^5y^7}{54x^2y^2}} = \frac{2xy\sqrt[3]{y^2}}{3}$: Simplify the fraction inside the radicand: cancel the shared factor of $2$ from the coefficients ($\frac{16}{54} = \frac{8}{27}$) and apply the Quotient Property for Exponents to the variables ($\frac{x^5}{x^2} = x^3$ and $\frac{y^7}{y^2} = y^5$). The expression becomes $\sqrt[3]{\frac{8x^3y^5}{27}}$. Rewrite using the Quotient Property: $\frac{\sqrt[3]{8x^3y^5}}{\sqrt[3]{27}}$. The denominator evaluates cleanly to $3$. Simplify the numerator by extracting perfect cube factors: $\sqrt[3]{8x^3y^3} \cdot \sqrt[3]{y^2} = 2xy\sqrt[3]{y^2}$. The simplified form is $\frac{2xy\sqrt[3]{y^2}}{3}$.

ⓒ $\sqrt[4]{\frac{5a^8b^6}{80a^3b^2}} = \frac{|ab|\sqrt[4]{a}}{2}$: Simplify the fraction inside the radicand: cancel the shared factor of $5$ ($\frac{5}{80} = \frac{1}{16}$) and simplify the variables ($\frac{a^8}{a^3} = a^5$ and $\frac{b^6}{b^2} = b^4$). The expression becomes $\sqrt[4]{\frac{a^5b^4}{16}}$. Rewrite using the Quotient Property: $\frac{\sqrt[4]{a^5b^4}}{\sqrt[4]{16}}$. The denominator evaluates to $2$. Simplify the numerator by extracting perfect fourth power factors: $\sqrt[4]{a^4b^4} \cdot \sqrt[4]{a}$. Because the index is even, taking the principal fourth root of $a^4b^4$ requires absolute value signs: $|ab|$. The simplified form is $\frac{|ab|\sqrt[4]{a}}{2}$.

These examples demonstrate that reducing the fraction under the radical first is crucial — it often reveals numerical and variable perfect powers that allow the expression to be evaluated fully without leaving radicals in the denominator.