\(\dfrac{\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}}=\dfrac{\left(\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}\right)^2}{\left(\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}\right)\left(\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}\right)}\)
\(=\dfrac{\left(\sqrt{2+\sqrt{3}}\right)^2+\left(\sqrt{2-\sqrt{3}}\right)^2+2\sqrt{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}}{\left(\sqrt{2+\sqrt{3}}\right)^2-\left(\sqrt{2-\sqrt{3}}\right)^2}\)
\(=\dfrac{2+\sqrt{3}+2-\sqrt{3}+2\sqrt{4-3}}{2+\sqrt{3}-2+\sqrt{3}}\)
\(=\dfrac{6}{2\sqrt{3}}=\sqrt{3}\)
