Simplify: $\;$ $\left[\dfrac{1}{\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right)^{-2}} - \left(\dfrac{\sqrt{a} - \sqrt{b}}{a^{\frac{3}{2}} - b^{\frac{3}{2}}}\right)^{-1}\right] \left(ab\right)^{\frac{-1}{2}}$
$\left[\dfrac{1}{\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right)^{-2}} - \left(\dfrac{\sqrt{a} - \sqrt{b}}{a^{\frac{3}{2}} - b^{\frac{3}{2}}}\right)^{-1}\right] \left(ab\right)^{\frac{-1}{2}}$
$= \left[\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right)^2 - \left(\dfrac{\left(a^{\frac{1}{2}}\right)^3 - \left(b^{\frac{1}{2}}\right)^3}{\sqrt{a} - \sqrt{b}}\right)\right] \times \dfrac{1}{\left(ab\right)^{\frac{1}{2}}}$
$= \left[\left(\sqrt{a} + \sqrt{b}\right)^2 - \dfrac{\left(\sqrt{a} - \sqrt{b}\right) \left(\left(\sqrt{a}\right)^2 + \sqrt{a} \sqrt{b} + \left(\sqrt{b}\right)^2\right)}{\sqrt{a} - \sqrt{b}} \right] \times \dfrac{1}{\sqrt{ab}}$
$= \left[a + b + 2 \sqrt{ab} - \left(a + \sqrt{ab} + b\right)\right] \times \dfrac{1}{\sqrt{ab}}$
$= \sqrt{ab} \times \dfrac{1}{\sqrt{ab}}$
$= 1$