Question

If $a=\sqrt{2}+1$, then find ${(a-\frac{1}{a})}^3$.

Answer 1

Given,

$a=\sqrt{2}+1$.

Now,

$\frac{1}{a}$

$=\frac{1}{\sqrt{2}+1}$

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

$=\sqrt{2}-1$.

Now,

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

So, ${(a-\frac{1}{a})}^3=2^3=8$.