$\displaystyle \begin{array}{{>{\displaystyle}l}} \sqrt{3+\sqrt{5}} =\sqrt{2}\sqrt{\frac{3}{2} +\frac{\sqrt{5}}{2}} =\sqrt{2}\sqrt{\frac{3}{2} +2.\frac{1}{2} .\frac{\sqrt{5}}{2}}\\ =\sqrt{2}\sqrt{\frac{1}{4} +2.\frac{1}{2} .\frac{\sqrt{5}}{2} +\frac{5}{4}} =\sqrt{2}\sqrt{\left(\frac{1}{2} +\frac{\sqrt{5}}{2}\right)^{2}}\\ =\sqrt{2}\left(\frac{1}{2} +\frac{\sqrt{5}}{2}\right) =\sqrt{\frac{1}{2}} +\sqrt{\frac{5}{2}} \end{array}$