1: \(2\sqrt{75}-5\sqrt{27}-\sqrt{192}+4\sqrt{48}\)
\(=2\cdot5\sqrt{3}-5\cdot3\sqrt{3}-8\sqrt{3}+4\cdot4\sqrt{3}\)
\(=10\sqrt{3}-15\sqrt{3}-8\sqrt{3}+16\sqrt{3}\)
\(=3\sqrt{3}\)
2: \(\dfrac{\sqrt{27}-3\sqrt{2}}{\sqrt{3}-\sqrt{2}}+\dfrac{6}{3+\sqrt{3}}+\dfrac{3}{\sqrt{3}}\)
\(=\dfrac{3\left(\sqrt{3}-\sqrt{2}\right)}{\sqrt{3}-\sqrt{2}}+\dfrac{6\left(3-\sqrt{3}\right)}{9-3}+\sqrt{3}\)
\(=3+3-\sqrt{3}+\sqrt{3}=6\)
3: \(\dfrac{2}{\sqrt{5}+1}+\sqrt{\dfrac{2}{3-\sqrt{5}}}\)
\(=\dfrac{2\left(\sqrt{5}-1\right)}{5-1}+\sqrt{\dfrac{2\left(3+\sqrt{5}\right)}{4}}\)
\(=\dfrac{\sqrt{5}-1}{2}+\sqrt{\dfrac{6+2\sqrt{5}}{4}}\)
\(=\dfrac{\sqrt{5}-1}{2}+\dfrac{\sqrt{5}+1}{2}=\dfrac{2\sqrt{5}}{2}=\sqrt{5}\)
4: \(\left(\dfrac{1}{3-\sqrt{5}}-\dfrac{1}{3+\sqrt{5}}\right)\cdot\dfrac{\sqrt{5}-1}{5-\sqrt{5}}\)
\(=\dfrac{3+\sqrt{5}-3+\sqrt{5}}{\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)}\cdot\dfrac{\sqrt{5}-1}{\sqrt{5}\left(\sqrt{5}-1\right)}\)
\(=\dfrac{2\sqrt{5}}{9-5}\cdot\dfrac{1}{\sqrt{5}}=\dfrac{2}{4}=\dfrac{1}{2}\)
