a: \(\dfrac{4\sqrt{3}}{\sqrt{60}}=4\cdot\sqrt{\dfrac{3}{60}}=\dfrac{4}{\sqrt{20}}=\dfrac{4}{2\sqrt{5}}=\dfrac{2}{\sqrt{5}}=\dfrac{2\sqrt{5}}{5}\)
b: \(\left(\sqrt{12}-2\sqrt{108}+\sqrt{675}\right):\sqrt{3}\)
\(=\left(2\sqrt{3}-2\cdot6\sqrt{3}+15\sqrt{3}\right):\sqrt{3}\)
\(=\left(17\sqrt{3}-12\sqrt{3}\right):\sqrt{3}=5\sqrt{3}:\sqrt{3}=5\)
c: \(\sqrt{6+\sqrt{20}}\cdot\sqrt{6-\sqrt{20}}\)
\(=\sqrt{\left(6+\sqrt{20}\right)\left(6-\sqrt{20}\right)}\)
\(=\sqrt{36-20}=\sqrt{16}=4\)
d: \(\left(\sqrt{6-\sqrt{11}}+\sqrt{6+\sqrt{11}}\right)^2\)
\(=6-\sqrt{11}+6+\sqrt{11}+2\cdot\sqrt{\left(6-\sqrt{11}\right)\left(6+\sqrt{11}\right)}\)
\(=12+2\cdot\sqrt{36-11}=12+2\cdot5=22\)
e: \(\dfrac{3\sqrt{10}+\sqrt{20}-3\sqrt{6}-\sqrt{12}}{\sqrt{5}-\sqrt{3}}\)
\(=\dfrac{\sqrt{10}\left(3+\sqrt{2}\right)-\sqrt{6}\left(3+\sqrt{2}\right)}{\sqrt{5}-\sqrt{3}}\)
\(=\dfrac{\left(3+\sqrt{2}\right)\left(\sqrt{10}-\sqrt{6}\right)}{\sqrt{5}-\sqrt{3}}\)
\(=\left(3+\sqrt{2}\right)\cdot\dfrac{\sqrt{2}\left(\sqrt{5}-\sqrt{3}\right)}{\sqrt{5}-\sqrt{3}}=\sqrt{2}\left(3+\sqrt{2}\right)=3\sqrt{2}+2\)
