\(\dfrac{3}{\sqrt{3}-1}=\dfrac{3\sqrt{3}+3}{2}\)
\(\dfrac{2+\sqrt{5}}{2-\sqrt{5}}=-9-4\sqrt{5}\)
\(\dfrac{4}{\sqrt{3}+1}=2\sqrt{3}-2\)
1) \(\dfrac{3}{\sqrt{3}-1}=\dfrac{3\left(\sqrt{3}+1\right)}{3-1}=\dfrac{3\sqrt{3}+3}{2}\)
2) \(=\dfrac{\left(2+\sqrt{5}\right)^2}{4-5}=-4-4\sqrt{5}-5=-9-4\sqrt{5}\)
3) \(=\dfrac{4\left(\sqrt{3}-1\right)}{3-1}=2\sqrt{3}-2\)
4) \(=\dfrac{2\sqrt{2}\left(\sqrt{5}+\sqrt{3}\right)}{5-3}=\sqrt{10}+\sqrt{6}\)
5) \(=\dfrac{2+2\sqrt{5}}{4-20}=-\dfrac{1+\sqrt{5}}{8}\)
6) \(=\dfrac{\left(2\sqrt{10}-5\right)\left(4+\sqrt{10}\right)}{16-10}=\dfrac{8\sqrt{10}+20-20-5\sqrt{10}}{4}=\dfrac{3\sqrt{10}}{4}\)
7) \(=\dfrac{2\sqrt{2}+3\sqrt{3}}{8-27}=-\dfrac{2\sqrt{2}+3\sqrt{3}}{19}\)
8) \(=\sqrt{\dfrac{\left(3-\sqrt{5}\right)^2}{9-5}}=\dfrac{3-\sqrt{5}}{2}\)
