Question

Simplify the following expressions.
$(\frac{a\sqrt{2}}{(1+a^2{)}^{-1}}-\frac{2\sqrt{2}}{a^{-1}})\frac{a^{-3}}{1-a^{-2}}$

Answer 1

$(\frac{a\sqrt{2}}{(1+a^2{)}^{-1}}-\frac{2\sqrt{2}}{a^{-1}})\frac{a^{-3}}{1-a^{-2}}$

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

$=\frac{a}{a^2}(\sqrt{2}+\sqrt{2}{a}^2-2\sqrt{2})\times \frac{1}{\frac{a^2-1}{a^2}}$

$=\frac{\sqrt{2}{a}^2-\sqrt{2}}{a^2-1}$

$=\sqrt{2}\frac{(a^2-1}{a^2-1}$

$=\sqrt{2}$