$\begin{align}  & \frac{5+\sqrt{7}}{3+\sqrt{7}}=\frac{5+\sqrt{7}}{3+\sqrt{7}}\times \frac{3-\sqrt{7}}{3-\sqrt{7}}=\frac{15-5\sqrt{7}+3\sqrt{7}-7}{9-7} \\ & \frac{5+\sqrt{7}}{3+\sqrt{7}}=\frac{8-2\sqrt{7}}{2}=4-\sqrt{7} \\\end{align}$