I have been asked to prove:

$$\sqrt{2+\sqrt3}-\sqrt{2-\sqrt3}=\sqrt2$$

Which I can easily do by converting the LHS to index form, then squaring it and simplifying it down to get 2, which is equal to the RHS squared, hence proved.

However I know you can't square a side during proof because it generates an extraneous solution. So: how do you go about this proof without squaring both sides? Or can my method be made valid if I do this: $$\sqrt{2+\sqrt3}-\sqrt{2-\sqrt3}=\sqrt2$$ $$...=...$$ $$2=2$$ $$\lvert\sqrt2\rvert=\lvert\sqrt2\rvert$$ $$\sqrt2=\sqrt2\text{ hence proved.}$$ Cheers in advance :)