[tex]\frac{\sqrt{5+\sqrt5}}{\sqrt{5-\sqrt5}}=\frac{\sqrt{5+\sqrt5}}{\sqrt{5-\sqrt5}}*\frac{\sqrt{5-\sqrt5}}{\sqrt{5-\sqrt5}}=\frac{\sqrt{(5+\sqrt5)(5-\sqrt5)}}{5-\sqrt5}=\frac{\sqrt{25-5}}{5-\sqrt5}=\frac{\sqrt{20}}{5-\sqrt5}=\frac{2\sqrt{5}}{5-\sqrt5}*\frac{5+\sqrt5}{5+\sqrt5}=\\=\frac{10\sqrt5+2*5}{25-5}=\frac{10\sqrt5+10}{20}=\frac{\sqrt5+1}{2}[/tex]
