Simplify: $\;$ $\left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\left(\dfrac{\sqrt[3]{y}}{y \sqrt{x}}\right)^{\frac{3}{2}} + \left(\dfrac{x^{\frac{-1}{2}}}{\sqrt[8]{y^3}}\right)^2\right]$
$\left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\left(\dfrac{\sqrt[3]{y}}{y \sqrt{x}}\right)^{\frac{3}{2}} + \left(\dfrac{x^{\frac{-1}{2}}}{\sqrt[8]{y^3}}\right)^2\right]$
$= \left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\dfrac{y^{\frac{1}{3} \times \frac{3}{2}}}{y^{\frac{3}{2}} x^{\frac{1}{2} \times \frac{3}{2}}} + \dfrac{x^{\frac{-1}{2} \times 2}}{y^{\frac{3}{8} \times 2}}\right]$
$= \left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\dfrac{y^{\frac{1}{2} - \frac{3}{2}}}{x^{\frac{3}{4}}} + \dfrac{x^{-1}}{y^{\frac{3}{4}}} \right]$
$= \left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\dfrac{y^{-1}}{x^{\frac{3}{4}}} + \dfrac{x^{-1}}{y^{\frac{3}{4}}} \right]$
$= \left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\dfrac{y^{-1 + \frac{3}{4}} + x^{-1 + \frac{3}{4}}}{x^{\frac{3}{4}} y^{\frac{3}{4}}}\right]$
$= \left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\dfrac{y^{\frac{-1}{4}} + x^{\frac{-1}{4}}}{x^{\frac{3}{4}} y^{\frac{3}{4}}}\right]$
$= \left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\dfrac{\dfrac{1}{y^{\frac{1}{4}}} + \dfrac{1}{x^{\frac{1}{4}}}}{x^{\frac{3}{4}} y^{\frac{3}{4}}}\right]$
$= \left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\dfrac{x^{\frac{1}{4}} + y^{\frac{1}{4}}}{x^{\frac{1}{4}} y^{\frac{1}{4}} \times x^{\frac{3}{4}} y^{\frac{3}{4}}}\right]$
$= \left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) \times \dfrac{xy}{\left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right)}$
$= xy$