\(E=\sqrt{\dfrac{5+2\sqrt{6}}{5-2\sqrt{6}}}+\sqrt{\dfrac{5-2\sqrt{6}}{5+2\sqrt{6}}}=\sqrt{\dfrac{\left(5+2\sqrt{6}\right)^2}{5^2-\left(2\sqrt{6}\right)^2}}+\sqrt{\dfrac{\left(5-2\sqrt{6}\right)^2}{5^2-\left(2\sqrt{6}\right)^2}}=\sqrt{\dfrac{\left(5+2\sqrt{6}\right)^2}{25-24}}+\sqrt{\dfrac{\left(5-2\sqrt{6}\right)^2}{25-24}}=\sqrt{\left(5+2\sqrt{6}\right)^2}+\sqrt{\left(5-2\sqrt{6}\right)^2}=5+2\sqrt{6}+5-2\sqrt{6}=5+5=10\)
\(E=\sqrt{\dfrac{5+2\sqrt{6}}{5-2\sqrt{6}}}+\sqrt{\dfrac{5-2\sqrt{6}}{5+2\sqrt{6}}}\)\(=\sqrt{\dfrac{\left(\sqrt{3}+\sqrt{2}\right)^2}{\left(\sqrt{3}-\sqrt{2}\right)^2}}+\sqrt{\dfrac{\left(\sqrt{3}-\sqrt{2}\right)^2}{\left(\sqrt{3}+\sqrt{2}\right)^2}}\)
\(=\sqrt{\left(\dfrac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}\right)^2}+\sqrt{\left(\dfrac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}\right)^2}\)
\(=\dfrac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}+\dfrac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}\)\(=\dfrac{\left(\sqrt{3}+\sqrt{2}\right)^2+\left(\sqrt{3}-\sqrt{2}\right)^2}{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)}\)
\(=\dfrac{3+2\sqrt{6}+2+2-2\sqrt{6}+3}{3-2}=10\)
