a) Ta có: \(\frac{\sqrt{15}-\sqrt{5}}{\sqrt{3}-1}-\frac{5-2\sqrt{5}}{2\sqrt{5}-4}\)
\(=\frac{\sqrt{5}\left(\sqrt{3}-1\right)}{\sqrt{3}-1}-\frac{\sqrt{5}\left(\sqrt{5}-2\right)}{2\left(\sqrt{5}-2\right)}\)
\(=\sqrt{5}-\frac{\sqrt{5}}{2}\)
\(=\frac{2\sqrt{5}-\sqrt{5}}{2}=\frac{\sqrt{5}}{2}\)
b) Ta có: \(\frac{2\sqrt{8}-\sqrt{12}}{\sqrt{3}-1}-\frac{\sqrt{5}+\sqrt{27}}{\sqrt{30}-\sqrt{2}}\)
\(=\frac{\left(2\sqrt{8}-\sqrt{12}\right)\left(\sqrt{3}+1\right)}{3-1}-\frac{\left(\sqrt{5}+\sqrt{27}\right)\left(\sqrt{30}+\sqrt{2}\right)}{30-2}\)
\(=\frac{4\sqrt{6}+4\sqrt{2}-6-2\sqrt{3}}{2}-\frac{5\sqrt{6}+\sqrt{10}+9\sqrt{10}+3\sqrt{6}}{28}\)
\(=\frac{4\sqrt{2}\left(\sqrt{3}+1\right)-2\sqrt{3}\left(\sqrt{3}+1\right)}{2}-\frac{10\sqrt{10}+8\sqrt{6}}{28}\)
\(=\frac{2\cdot\left(\sqrt{3}+1\right)\left(2\sqrt{2}-\sqrt{3}\right)}{2}-\frac{\sqrt{1000}+\sqrt{384}}{28}\)
\(=2\sqrt{6}-3+2\sqrt{2}-\sqrt{3}-\frac{\sqrt{2}\cdot\left(5\sqrt{5}+4\sqrt{3}\right)}{14}\)
\(=\frac{28\sqrt{6}-42+28\sqrt{2}-14\sqrt{3}-10\sqrt{10}-8\sqrt{6}}{14}\)
\(=\frac{20\sqrt{6}-42+28\sqrt{2}-14\sqrt{3}-10\sqrt{10}}{14}\)
c) Ta có: \(\frac{2}{\sqrt{3}-1}-\frac{2}{\sqrt{3}+1}\)
\(=\frac{2\left(\sqrt{3}+1\right)-2\left(\sqrt{3}-1\right)}{3-1}\)
\(=\frac{2\sqrt{3}+2-2\sqrt{3}+2}{2}\)
\(=\frac{4}{2}=2\)
