Question

1/2+root 3 +2/root5-root3 +1/2- root 5=0

Answer 1

Dear student
$$\begin{gathered} \frac{1}{2+\sqrt{3}}+\frac{2}{\sqrt{5}-\sqrt{3}}+\frac{1}{2-\sqrt{5}} \\ =\frac{1}{2+\sqrt{3}}\times \frac{2-\sqrt{3}}{2-\sqrt{3}}+\frac{2}{\sqrt{5}-\sqrt{3}}\times \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}+\frac{1}{2-\sqrt{5}}\times \frac{2+\sqrt{5}}{2+\sqrt{5}} \\ =\frac{2-\sqrt{3}}{4-3}+\frac{2\left(\sqrt{5}+\sqrt{3}\right)}{5-3}+\frac{2+\sqrt{5}}{4-5} \\ =2-\sqrt{3}+\frac{2\sqrt{5}+2\sqrt{3}}{2}-\left(2+\sqrt{5}\right) \\ =\frac{4-2\sqrt{3}+2\sqrt{5}+2\sqrt{3}-4-2\sqrt{5}}{2} \\ =0 \end{gathered}$$
Regards