1: \(\dfrac{6}{\sqrt{7}-1}+\dfrac{1}{\sqrt{2}+1}+\dfrac{9}{4+\sqrt{7}}\)
\(=\dfrac{6\left(\sqrt{7}+1\right)}{7-1}+\dfrac{1\left(\sqrt{2}-1\right)}{2-1}+\dfrac{9\left(4-\sqrt{7}\right)}{16-7}\)
\(=\sqrt{7}+1+\sqrt{2}-1+4-\sqrt{7}\)
\(=4+\sqrt{2}\)
2: \(\dfrac{2}{\sqrt{13}-\sqrt{11}}+\dfrac{10}{\sqrt{11}-1}+\dfrac{9}{\sqrt{13}-2}\)
\(=\dfrac{2\left(\sqrt{13}+\sqrt[]{11}\right)}{13-11}+\dfrac{10\left(\sqrt{11}+1\right)}{11-1}+\dfrac{9\left(\sqrt{13}+2\right)}{13-4}\)
\(=\sqrt{13}+\sqrt{11}+\sqrt{11}+1+\sqrt{13}+2=2\sqrt{13}+2\sqrt{11}+3\)
3: \(\dfrac{2}{\sqrt{7}-\sqrt{5}}+\dfrac{6}{\sqrt{11}+\sqrt{5}}\)
\(=\dfrac{2\left(\sqrt{7}+\sqrt{5}\right)}{7-5}+\dfrac{6\left(\sqrt{11}-\sqrt{5}\right)}{11-5}\)
\(=\sqrt{7}+\sqrt{5}+\sqrt{11}-\sqrt{5}=\sqrt{11}+\sqrt{7}\)
4: \(\dfrac{\sqrt{15}-\sqrt{12}}{\sqrt{5}-2}-\dfrac{1}{2-\sqrt{3}}\)
\(=\dfrac{\sqrt{3}\left(\sqrt{5}-2\right)}{\sqrt{5}-2}-\dfrac{2+\sqrt{3}}{4-3}\)
\(=\sqrt{3}-2-\sqrt{3}\)
=-2
5: \(\dfrac{3}{\sqrt{5}+\sqrt{2}}+\dfrac{1}{\sqrt{2}-1}-\dfrac{4}{3-\sqrt{5}}\)
\(=\dfrac{3\left(\sqrt{5}-\sqrt{2}\right)}{5-2}+\dfrac{1\left(\sqrt{2}+1\right)}{2-1}-\dfrac{4\left(3+\sqrt{5}\right)}{4}\)
\(=\sqrt{5}-\sqrt{2}+\sqrt{2}+1-3-\sqrt{5}=-2\)
6: \(\dfrac{1}{2+\sqrt{3}}+\dfrac{\sqrt{2}}{\sqrt{6}}-\dfrac{2}{3+\sqrt{3}}\)
\(=\dfrac{2-\sqrt{3}}{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}+\dfrac{1}{\sqrt{3}}-\dfrac{2}{3+\sqrt{3}}\)
\(=2-\sqrt{3}+\dfrac{\sqrt{3}+1-2}{3+\sqrt{3}}\)
\(=\dfrac{\left(2-\sqrt{3}\right)\left(3+\sqrt{3}\right)+\sqrt{3}-1}{3+\sqrt{3}}\)
\(=\dfrac{6+2\sqrt{3}-3\sqrt{3}-3+\sqrt{3}-1}{3+\sqrt{3}}=\dfrac{2}{3+\sqrt{3}}\)
\(=\dfrac{2\left(3-\sqrt{3}\right)}{9-3}=\dfrac{2}{6}\cdot\left(3-\sqrt{3}\right)=\dfrac{3-\sqrt{3}}{3}\)
