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