Evaluating the expression: $$\sqrt{5+2\sqrt{6}}+ \sqrt{5-2\sqrt{6}} $$
So my solution involves rewriting the expression inside the bigger radical signs as perfect squares: $$\sqrt{\left(\sqrt{3}+\sqrt{2} \right)^2}+ \sqrt{\left(\sqrt{3}-\sqrt{2} \right)^2} $$
From which I then get $\sqrt{3}+\sqrt{2} + \sqrt{3}-\sqrt{2} = 2\sqrt{3} $. However we can also write the original expression as $$\sqrt{\left(\sqrt{2}+\sqrt{3} \right)^2}+ \sqrt{\left(\sqrt{2}-\sqrt{3} \right)^2} $$ and from this get $\sqrt{2}+\sqrt{3} + \sqrt{2}-\sqrt{3} = 2\sqrt{2} $.
What am I missing or doing wrong here?