Simplify: $\;$ $\left(\dfrac{2a + b^{\frac{1}{2}} a^{\frac{1}{2}}}{3a}\right)^{-1} \left(\dfrac{a^{\frac{3}{2}} - b^{\frac{3}{2}}}{a - a^{\frac{1}{2}} b^{\frac{1}{2}}} - \dfrac{a - b}{\sqrt{a} + \sqrt{b}} \right)$
Given expression $\;\;$ $\left(\dfrac{2a + b^{\frac{1}{2}} a^{\frac{1}{2}}}{3a}\right)^{-1} \left(\dfrac{a^{\frac{3}{2}} - b^{\frac{3}{2}}}{a - a^{\frac{1}{2}} b^{\frac{1}{2}}} - \dfrac{a - b}{\sqrt{a} + \sqrt{b}} \right)$ $\;\;\; \cdots \; (1)$
Consider the expression $\;\;$ $\left(\dfrac{2a + b^{\frac{1}{2}} a^{\frac{1}{2}}}{3a}\right)^{-1}$
$= \dfrac{3a}{2a + b^{\frac{1}{2}} a^{\frac{1}{2}}}$
$= \dfrac{3 \sqrt{a} \times \sqrt{a}}{2a + \sqrt{a} \sqrt{b}}$
$= \dfrac{3 \sqrt{a} \times \sqrt{a}}{\sqrt{a} \left(2 \sqrt{a} + \sqrt{b}\right)}$
$= \dfrac{3 \sqrt{a}}{2 \sqrt{a} + \sqrt{b}}$ $\;\;\; \cdots \; (2)$
Consider the expression $\;\;$ $\dfrac{a^{\frac{3}{2}} - b^{\frac{3}{2}}}{a - a^{\frac{1}{2}} b^{\frac{1}{2}}} - \dfrac{a - b}{\sqrt{a} + \sqrt{b}}$
$= \dfrac{\left(a^{\frac{1}{2}}\right)^3 - \left(b^{\frac{1}{2}}\right)^3}{a - \sqrt{a} \sqrt{b}} - \dfrac{\left(a - b\right) \left(\sqrt{a} - \sqrt{b}\right)}{\left(\sqrt{a} + \sqrt{b}\right) \left(\sqrt{a} - \sqrt{b}\right)}$
$= \dfrac{\left(\sqrt{a} - \sqrt{b}\right) \left(a + \sqrt{ab} + b\right)}{\sqrt{a} \left(\sqrt{a} - \sqrt{b}\right)} - \dfrac{\left(a - b\right) \left(\sqrt{a} - \sqrt{b}\right)}{a - b}$
$= \dfrac{a + \sqrt{ab} + b}{\sqrt{a}} - \left(\sqrt{a} - \sqrt{b}\right)$
$= \dfrac{a + \sqrt{ab} + b - a + \sqrt{ab}}{\sqrt{a}}$
$= \dfrac{2 \sqrt{ab} + b}{\sqrt{a}}$
$= \dfrac{\sqrt{b} \left(2 \sqrt{a} + \sqrt{b}\right)}{\sqrt{a}}$ $\;\;\; \cdots \; (3)$
In view of expressions $(2)$ and $(3)$, expression $(1)$ becomes
$\dfrac{3 \sqrt{a}}{2 \sqrt{a} + \sqrt{b}} \times \dfrac{\sqrt{b} \left(2 \sqrt{a} + \sqrt{b}\right)}{\sqrt{a}}$
$= 3 \sqrt{b}$