ルート引くルートの問題

2乗すると、
\begin{align*} \left(\sqrt{2\sqrt{3}+2}-\sqrt{\sqrt{3}-\sqrt{2}}\,\right)^{2} & =\left(2\sqrt{3}+2\right)+\left(\sqrt{3}-\sqrt{2}\right)-2\sqrt{2\sqrt{3}+2}\sqrt{\sqrt{3}-\sqrt{2}}\\ & =3\sqrt{3}-\sqrt{2}+2-2\sqrt{6-2\sqrt{6}+2\sqrt{3}-2\sqrt{2}}\\ & =3\sqrt{3}-\sqrt{2}+2-2\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^{2}+1+2\left(\sqrt{3}-\sqrt{2}\right)}\\ & =3\sqrt{3}-\sqrt{2}+2-2\sqrt{\left(\sqrt{3}-\sqrt{2}+1\right)^{2}}\\ & =3\sqrt{3}-\sqrt{2}+2-2\left|\sqrt{3}-\sqrt{2}+1\right|\\ & =3\sqrt{3}-\sqrt{2}+2-2\left(\sqrt{3}-\sqrt{2}+1\right)\\ & =\sqrt{3}+\sqrt{2} \end{align*} となり、\(0<\sqrt{2\sqrt{3}+2}-\sqrt{\sqrt{3}-\sqrt{2}}\)なので、
\[ \sqrt{2\sqrt{3}+2}-\sqrt{\sqrt{3}-\sqrt{2}}=\sqrt{\sqrt{3}+\sqrt{2}} \]