Prove that $\sqrt{7-\sqrt{2}} $ is irrational My idea is the following
$\sqrt{7-\sqrt{2}} \in\mathbb{Q}$
$w^2=7+\sqrt{2}$
$w^2-7=\sqrt{2}$ thus w cannot be rational is this correct?
edit: or maybe i can assume $\sqrt{7-\sqrt{2}} =\frac{m}{n}$ and then square both sides but the i am not sure what to do next
We can even show $7-\sqrt2$ is irrational
as if $7-\sqrt2=a$ is rational
$$7^2+2-14\sqrt2=a^2\iff \sqrt2=(51-a^2)/(14a)$$ which is rational unlike $\sqrt2$