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