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