Question

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

Answer 1

$a=1-\sqrt{2}$

$\Rightarrow \frac{1}{a}=\frac{1}{1-\sqrt{2}}\times \frac{1+\sqrt{2}}{1+\sqrt{2}}=\frac{1+\sqrt{2}}{1-2}=-1-\sqrt{2}$

Substituting in ${(a-\frac{1}{a})}^2={(1-\sqrt{2}+1+\sqrt{2})}^2=2^2=4$