Simplify: $\;$ $\left(a^2 \sqrt{b}\right)^{\frac{-1}{2}} \left(\sqrt{ab} - \dfrac{ab}{a + \sqrt{ab}}\right) : \dfrac{\sqrt[4]{ab} - \sqrt{b}}{a - b}$
$\left(a^2 \sqrt{b}\right)^{\frac{-1}{2}} \left(\sqrt{ab} - \dfrac{ab}{a + \sqrt{ab}}\right) : \dfrac{\sqrt[4]{ab} - \sqrt{b}}{a - b}$
$= \left(\dfrac{1}{\left(a^2\right)^{\frac{1}{2}} \left(b^{\frac{1}{2}}\right)^{\frac{1}{2}}} \times \dfrac{a \sqrt{ab} + ab - ab}{a + \sqrt{ab}} \right) : \dfrac{\sqrt[4]{ab} - \sqrt{b}}{a - b}$
$= \dfrac{1}{a b^{\frac{1}{4}}} \times \dfrac{a \times a^{\frac{1}{2}} b^{\frac{1}{2}}}{a + \sqrt{ab}} \times \dfrac{a - b}{\sqrt[4]{ab} - \sqrt{b}}$
$= \dfrac{a^{\frac{1}{2}} b^{\frac{1}{4}}}{a + a^{\frac{1}{2}} b^{\frac{1}{2}}} \times \dfrac{a - b}{a^{\frac{1}{4}} b^{\frac{1}{4}} - b^{\frac{1}{2}}}$
$= \dfrac{a^{\frac{1}{2}} b^{\frac{1}{4}}}{a^{\frac{1}{2}} \left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right)} \times \dfrac{a - b}{b^{\frac{1}{4}} \left(a^{\frac{1}{4}} - b^{\frac{1}{4}}\right)}$
$= \dfrac{a - b}{\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right) \left(a^{\frac{1}{4}} - b^{\frac{1}{4}}\right)}$
$= \dfrac{\left(a - b\right) \left(a^{\frac{1}{4}} + b^{\frac{1}{4}}\right)}{\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right) \left(a^{\frac{1}{4}} - b^{\frac{1}{4}}\right) \left(a^{\frac{1}{4}} + b^{\frac{1}{4}}\right)}$
$= \dfrac{\left(a - b\right) \left(a^{\frac{1}{4}} + b^{\frac{1}{4}}\right)}{\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right) \left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right)}$
$= \dfrac{\left(a - b\right) \left(a^{\frac{1}{4}} + b^{\frac{1}{4}}\right)}{a - b}$
$= a^{\frac{1}{4}} + b^{\frac{1}{4}}$