TITLE: Finding the square root of $6-4\sqrt{2}$
QUESTION [6 upvotes]: I found this standupmaths video on YouTube about the A4 paper puzzle.
I really liked the puzzle and managed to get the answer by using a calculator. However, the answer (which I won't spoil), led me to think that the equation to solve it might simplify - which it does.
In the middle of the simplification, I got this expression:
$$\sqrt{6-4\sqrt{2}}$$
which for other reasons I suspected to be equal to:
$$\ 2-\sqrt{2}$$
I tried squaring the above and, sure enough, it does give:
$$6-4\sqrt{2}$$
My question is, how would I have been able to find the square root of
$$6-4\sqrt{2}$$
if I hadn't been able to guess at it? Is there a standard technique? I've tried looking on the web but don't really even know what to search for.

REPLY [2 votes]: We have to assume that the nested radical can be rewritten as the sum of two other radicals (surds).
$\sqrt{6-4\sqrt{2}}=\sqrt{d}+\sqrt{e}$
Squaring both sides gives us $$6-4\sqrt{2}=d+e+2\sqrt{de}$$
This can be solved by finding $2$ numbers that sum to $6$ and multiply to $4^2\cdot 2/4=8$
($2\sqrt{de}=4\sqrt{2}\rightarrow de=8$)
Numbers $4$ and $2$ work, so $$\sqrt{6-4\sqrt{2}}=\sqrt{4}-\sqrt{2}=2-\sqrt{2}$$