Algebraic simplification including square roots $\sqrt{2}\sqrt{3 + 2\sqrt{2}} $ This may be stupid, but how do I see that $$\sqrt{2}\sqrt{3 + 2\sqrt{2}} - 1 = 1 + \sqrt{2}$$
having only the left-hand side? 

You may note that $3+2\sqrt{2}=(1+\sqrt{2})^2$. Hence, we have
$$
\sqrt{2}\sqrt{3+2\sqrt{2}}-1=\sqrt{2}(1+\sqrt{2})-1=\sqrt{2}+2-1=\sqrt{2}+1,
$$
as desired.
