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