a: \(\dfrac{\sqrt{7}-5}{2}-\dfrac{6-2\sqrt{7}}{4}+\dfrac{6}{\sqrt{7}-2}-\dfrac{5}{4+\sqrt{7}}\)
\(=\dfrac{\sqrt{7}-5-\left(3-\sqrt{7}\right)}{2}+\dfrac{6\left(\sqrt{7}+2\right)}{3}-\dfrac{5\left(4-\sqrt{7}\right)}{9}\)
\(=\dfrac{-8+2\sqrt{7}}{2}+\dfrac{18\left(\sqrt{7}+2\right)-5\left(4-\sqrt{7}\right)}{9}\)
\(=-4+\sqrt{7}+\dfrac{23\sqrt{7}+16}{9}\)
\(=\dfrac{-36+9\sqrt{7}+23\sqrt{7}+16}{9}=\dfrac{-20+32\sqrt{7}}{9}\)
b:
\(\dfrac{2}{\sqrt{6}-2}+\dfrac{2}{\sqrt{6}+2}+\dfrac{5}{\sqrt{6}}\)
\(=\dfrac{2\left(\sqrt{6}+2\right)+2\left(\sqrt{6}-2\right)}{2}+\dfrac{5\sqrt{6}}{6}\)
\(=2\sqrt{6}+\dfrac{5}{6}\sqrt{6}=\dfrac{17}{6}\sqrt{6}\)
c:
\(\dfrac{1}{\sqrt{3}+\sqrt{2}-\sqrt{5}}-\dfrac{1}{\sqrt{3}+\sqrt{2}+\sqrt{5}}\)
\(=\dfrac{\sqrt{3}+\sqrt{2}+\sqrt{5}-\sqrt{3}-\sqrt{2}+\sqrt{5}}{5+2\sqrt{6}-5}\)
\(=\dfrac{2\sqrt{5}}{2\sqrt{6}}=\dfrac{\sqrt{30}}{6}\)
d:
\(\left(\dfrac{\sqrt{6}-\sqrt{2}}{1-\sqrt{3}}-\dfrac{5}{\sqrt{5}}\right):\dfrac{1}{\sqrt{5}-\sqrt{2}}\)
\(=\left(\dfrac{\sqrt{2}\left(\sqrt{3}-1\right)}{-\left(\sqrt{3}-1\right)}-\sqrt{5}\right)\cdot\left(\sqrt{5}-\sqrt{2}\right)\)
\(=-\left(\sqrt{5}+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{2}\right)\)
=-(5-2)
=-3
e:
\(\dfrac{1}{\sqrt{3}}+\dfrac{1}{3\sqrt{2}}+\dfrac{1}{\sqrt{3}}\cdot\sqrt{\dfrac{5}{12}-\dfrac{1}{\sqrt{6}}}\)
\(=\dfrac{\sqrt{6}+1}{3\sqrt{2}}+\dfrac{1}{\sqrt{3}}\cdot\sqrt{\dfrac{5-2\sqrt{6}}{12}}\)
\(=\dfrac{2\sqrt{3}+\sqrt{2}}{6}+\dfrac{1}{\sqrt{3}}\cdot\dfrac{\sqrt{3}-\sqrt{2}}{2\sqrt{3}}\)
\(=\dfrac{2\sqrt{3}+\sqrt{2}+\sqrt{3}-\sqrt{2}}{6}=\dfrac{3\sqrt{3}}{6}=\dfrac{\sqrt{3}}{2}\)
\(a,\dfrac{\sqrt{7}-5}{2}-\dfrac{6-2\sqrt{7}}{4}+\dfrac{6}{\sqrt{7}-2}-\dfrac{5}{4+\sqrt{7}}\)
\(=\dfrac{2\sqrt{7}-10-6+2\sqrt{7}}{4}+\dfrac{6\sqrt{7}+12}{3}-\dfrac{20-5\sqrt{7}}{9}\)
\(=\dfrac{4\sqrt{7}-16}{4}+\dfrac{6\sqrt{7}+12}{3}-\dfrac{20-5\sqrt{7}}{9}\)
\(=\sqrt{7}-4+2\sqrt{7}+4-\dfrac{20-5\sqrt{7}}{9}\)
\(=\dfrac{9\sqrt{7}-36+18\sqrt{7}+36-20+5\sqrt{7}}{9}\)
\(=\dfrac{32\sqrt{7}-20}{9}\)
\(b,\dfrac{2}{\sqrt{6}-2}+\dfrac{2}{\sqrt{6}+2}+\dfrac{5}{\sqrt{6}}\)
\(=\dfrac{2\left(\sqrt{6}+2\right)}{2}+\dfrac{2\left(\sqrt{6}-2\right)}{2}+\dfrac{5}{\sqrt{6}}\)
\(=\dfrac{4\sqrt{6}}{2}+\dfrac{5}{\sqrt{6}}\)
\(=\dfrac{17\sqrt{6}}{6}\)
\(c,\dfrac{1}{\sqrt{3}+\sqrt{2}-\sqrt{5}}-\dfrac{1}{\sqrt{3}+\sqrt{2}+\sqrt{5}}\)
\(=\dfrac{\sqrt{3}+\sqrt{2}+\sqrt{5}}{3+2+2\sqrt{6}-5}-\dfrac{\sqrt{3}+\sqrt{2}-\sqrt{5}}{3+2+2\sqrt{6}-5}\)
\(=\dfrac{\sqrt{3}+\sqrt{2}+\sqrt{5}-\sqrt{3}-\sqrt{2}+\sqrt{5}}{2\sqrt{6}}\)
\(=\dfrac{2\sqrt{5}}{2\sqrt{6}}\)
\(=\dfrac{\sqrt{30}}{6}\)
