\(\dfrac{2-\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\dfrac{2+\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}\)
\(=\dfrac{\sqrt{2}\left(2-\sqrt{3}\right)}{2+\sqrt{4+2\sqrt{3}}}+\dfrac{\sqrt{2}\left(2+\sqrt{3}\right)}{2-\sqrt{4-2\sqrt{3}}}\)
\(=\sqrt{2}.\left(\dfrac{2-\sqrt{3}}{2+\sqrt{\left(\sqrt{3}+1\right)^2}}+\dfrac{2+\sqrt{3}}{2-\sqrt{\left(\sqrt{3}-1\right)^2}}\right)\)
\(=\sqrt{2}.\left(\dfrac{2-\sqrt{3}}{2+\left|\sqrt{3}+1\right|}+\dfrac{2+\sqrt{3}}{2-\left|\sqrt{3}-1\right|}\right)\)
\(=\sqrt{2}.\left(\dfrac{2-\sqrt{3}}{3+\sqrt{3}}+\dfrac{2+\sqrt{3}}{3-\sqrt{3}}\right)\)
\(=\sqrt{2}.\left[\dfrac{2-\sqrt{3}}{\sqrt{3}\left(1+\sqrt{3}\right)}+\dfrac{2+\sqrt{3}}{\sqrt{3}\left(\sqrt{3}-1\right)}\right]\)
\(=\sqrt{\dfrac{2}{3}}.\left(\dfrac{2-\sqrt{3}}{\sqrt{3}+1}+\dfrac{2+\sqrt{3}}{\sqrt{3}-1}\right)\)
\(=\sqrt{\dfrac{2}{3}}.\left[\dfrac{\left(2-\sqrt{3}\right)\left(\sqrt{3}-1\right)+\left(2+\sqrt{3}\right)\left(\sqrt{3}+1\right)}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}\right]\)
\(=\sqrt{\dfrac{2}{3}}.\left(\dfrac{2\sqrt{3}-2-3+\sqrt{3}+2\sqrt{3}+2+3+\sqrt{3}}{3-1}\right)\)
\(=\sqrt{\dfrac{2}{3}}.\left(\dfrac{6\sqrt{3}}{2}\right)\)
\(=\sqrt{\dfrac{2}{3}}.3\sqrt{3}=3\sqrt{2}\)
