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