Simplify: $\;$ $\left(2^{\frac{3}{2}} + 27 y^{\frac{3}{5}}\right) : \left[\left(\dfrac{1}{2}\right)^{\frac{-1}{2}} + 3 y^{\frac{1}{5}}\right]$
$\left(2^{\frac{3}{2}} + 27 y^{\frac{3}{5}}\right) : \left[\left(\dfrac{1}{2}\right)^{\frac{-1}{2}} + 3 y^{\frac{1}{5}}\right]$
$= \dfrac{\left(2^{\frac{1}{2}}\right)^3 + \left(3 y^{\frac{1}{5}}\right)^3}{2^{\frac{1}{2}} + 3 y^{\frac{1}{5}}}$
$= \dfrac{\left(2^{\frac{1}{2}} + 3y^{\frac{1}{5}}\right) \left[\left(2^{\frac{1}{2}}\right)^2 - 2^{\frac{1}{2}} \times 3 y^{\frac{1}{5}} + \left(3y^{\frac{1}{5}}\right)^2 \right]}{2^{\frac{1}{2}} + 3 y^{\frac{1}{5}}}$
$= 2 - 3 \sqrt{2} y^{\frac{1}{5}} + 9 y^{\frac{2}{5}}$