\(\sqrt[]{3+\sqrt{5}}+\sqrt{3-\sqrt{5}}=\sqrt{\left(\sqrt{\dfrac{5}{2}}+\sqrt{\dfrac{1}{2}}\right)^2}+\sqrt{\left(\sqrt{\dfrac{5}{2}}-\sqrt{\dfrac{1}{2}}\right)^2}=\sqrt{\dfrac{5}{2}}+\sqrt{\dfrac{1}{2}}+\sqrt{\dfrac{5}{2}}-\sqrt{\dfrac{1}{2}}=\sqrt{10}\)
\(\sqrt{3+\sqrt{5}}+\sqrt{3-\sqrt{5}}\)
= \(\dfrac{\sqrt{2}.\sqrt{3+\sqrt{5}}}{\sqrt{2}}+\dfrac{\sqrt{2}.\sqrt{3-\sqrt{5}}}{\sqrt{2}}\)
= \(\dfrac{\sqrt{6+2\sqrt{5}}}{\sqrt{2}}+\dfrac{\sqrt{6-2\sqrt{5}}}{\sqrt{2}}\)
=\(\dfrac{\sqrt{\left(\sqrt{5}+1\right)}}{\sqrt{2}}+\dfrac{\sqrt{\left(\sqrt{5}-1\right)}}{\sqrt{2}}\)
= I \(\sqrt{5}+1\)I / \(\sqrt{2}\)+ I\(\sqrt{5}-1\)I/\(\sqrt{2}\)
= \(\dfrac{\sqrt{5}+1}{\sqrt{2}}+\dfrac{\sqrt{5}-1}{\sqrt{2}}\)
=\(\dfrac{\sqrt{5}+1+\sqrt{5}-1}{\sqrt{2}}\)
= \(\dfrac{\sqrt{5}}{\sqrt{2}}\)
= \(\dfrac{\sqrt{5}.\sqrt{2}}{\sqrt{2}.\sqrt{2}}\)
= \(\dfrac{\sqrt{10}}{2}\)
