Answer
$\frac{8x^{\frac{5}{4}}}{y^{\frac{3}{4}}}$
Work Step by Step
Use $16=2^4$.
$\dfrac{(16x^2y^{-\frac{1}{3}})^{\frac{3}{4}}}{(xy^2)^{\frac{1}{4}}}=\dfrac{(2^4x^2y^{-\frac{1}{3}})^{\frac{3}{4}}}{(xy^2)^{\frac{1}{4}}}$
Use the rule $\left(ab\right)^m=a^mb^m$:
$=\dfrac{\left(2^4\right)^{\frac{3}{4}}\left(x^2\right)^{\frac{3}{4}}\left(y^{-\frac{1}{3}}\right)^{\frac{3}{4}}}{x^{\frac{1}{4}}\left(y^2\right)^{\frac{1}{4}}}$
Use the rule $\left(a^m\right)^n=a^{mn}$, then simplify to obtain:
$=\dfrac{2^{4\cdot \frac{3}{4}}x^{2\cdot \frac{3}{4}}y^{-\frac{1}{3}\cdot \frac{3}{4}}}{x^{\frac{1}{4}}y^{2\cdot \frac{1}{4}}}$
$=\dfrac{2^3x^{ \frac{3}{2}}y^{-\frac{1}{4}}}{x^{\frac{1}{4}}y^{\frac{1}{2}}}$
$=\dfrac{8x^{ \frac{3}{2}}y^{-\frac{1}{4}}}{x^{\frac{1}{4}}y^{\frac{1}{2}}}$
Use the rule $\dfrac{a^m}{a^n} = a^{m-n}$:
$=8x^{\frac{3}{2}-\frac{1}{4}}y^{-\frac{1}{4}-\frac{1}{2}}$
$=8x^{\frac{6}{4}-\frac{1}{4}}y^{-\frac{1}{4}-\frac{2}{4}}$
$=8x^{\frac{5}{4}}y^{-\frac{3}{4}}$
Use the rule $a^{-m}=\dfrac{1}{a^m}$, then simplify to obtain:
$=8x^{\frac{5}{4}}\cdot \dfrac{1}{y^{\frac{3}{4}}}$
$=\dfrac{8x^{\frac{5}{4}}}{y^{\frac{3}{4}}}$
Hence, the correct answer is $\frac{8x^{\frac{5}{4}}}{y^{\frac{3}{4}}}$.
