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