a: \(\frac{5}{\sqrt3+1}-\frac{5}{\sqrt3-1}\)
\(=\frac{5\left(\sqrt3-1\right)-5\left(\sqrt3+1\right)}{\left(\sqrt3+1\right)\left(\sqrt3-1\right)}\)
\(=\frac{5\sqrt3-5-5\sqrt3-5}{3-1}=\frac{-10}{2}=-5\)
b: \(\frac{7}{\sqrt5-\sqrt3}+\frac{7}{\sqrt5+\sqrt3}\)
\(=\frac{7\left(\sqrt5+\sqrt3\right)+7\left(\sqrt5-\sqrt3\right)}{\left(\sqrt5+\sqrt3\right)\left(\sqrt5-\sqrt3\right)}\)
\(=\frac{7\sqrt5+7\sqrt3+7\sqrt5-7\sqrt3}{5-3}=\frac{14\sqrt5}{2}=7\sqrt5\)
c: \(\frac{\sqrt5+1}{\sqrt5-1}-\frac{\sqrt5-1}{\sqrt5+1}\)
\(=\frac{\left(\sqrt5+1\right)^2-\left(\sqrt5-1\right)^2}{\left(\sqrt5-1\right)\left(\sqrt5+1\right)}\)
\(=\frac{6+2\sqrt5-\left(6-2\sqrt5\right)}{5-1}=\frac{4\sqrt5}{4}=\sqrt5\)
d: \(\frac{\sqrt7-\sqrt2}{\sqrt7+\sqrt2}+\frac{\sqrt7+\sqrt2}{\sqrt7-\sqrt2}\)
\(=\frac{\left(\sqrt7-\sqrt2\right)^2+\left(\sqrt7+\sqrt2\right)^2}{\left(\sqrt7-\sqrt2\right)\left(\sqrt7+\sqrt2\right)}\)
\(=\frac{9-2\sqrt{14}+9+2\sqrt{14}}{7-2}=\frac{18}{5}\)
