Question

If $a=\sqrt{2}+1$, then find the value of $a-\frac{1}{a}$.

Answer 1

$a=\sqrt{2}+1$

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

(Rationalising the denominator)

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

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

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