a: \(\left(\frac{2}{\sqrt3-1}+\frac{3}{\sqrt3-2}+\frac{15}{3-\sqrt3}\right)\cdot\frac{1}{5+\sqrt3}\)
\(=\left\lbrack\frac{2\left(\sqrt3+1\right)}{\left(\sqrt3-1\right)\left(\sqrt3+1\right)}-\frac{3\left(2+\sqrt3\right)}{\left(2-\sqrt3\right)\left(2+\sqrt3\right)}+\frac{15\left(3+\sqrt3\right)}{\left(3-\sqrt3\right)\left(3+\sqrt3\right)}\right\rbrack\cdot\frac{1}{5+\sqrt3}\)
\(=\left\lbrack\sqrt3+1-6-3\sqrt3+\frac52\left(3+\sqrt3\right)\right\rbrack\cdot\frac{1}{5+\sqrt3}=\left(-2\sqrt3-5+\frac{15}{2}+\frac52\sqrt3\right)\cdot\frac{1}{5+\sqrt3}\)
\(=\left(\frac52+\frac12\cdot\sqrt3\right)\cdot\frac{1}{5+\sqrt3}=\frac12\left(5+\sqrt3\right)\cdot\frac{1}{5+\sqrt3}=\frac12\)
b: \(\frac{\sqrt3}{\sqrt{\sqrt3+1}-1}-\frac{\sqrt3}{\sqrt{\sqrt3+1}+1}\)
\(=\frac{\sqrt3\left(\sqrt{\sqrt3+1}+1\right)-\sqrt3\left(\sqrt{\sqrt3+1}-1\right)}{\sqrt3+1-1}=\frac{\sqrt3\left(\sqrt{\sqrt3+1}+1-\sqrt{\sqrt3+1}+1\right)}{\sqrt3}=2\)
c: \(\sqrt{\frac{3}{20}}+\sqrt{\frac{1}{60}}-2\cdot\sqrt{\frac{1}{15}}\)
\(=\frac{\sqrt3}{2\sqrt5}+\frac{1}{2\sqrt{15}}-\frac{2}{\sqrt{15}}=\frac{3}{2\sqrt{15}}+\frac{1}{2\sqrt{15}}-\frac{4}{2\sqrt{15}}\)
=0
d: \(\left(\frac{\sqrt{14}-\sqrt7}{1-\sqrt2}+\frac{\sqrt{15}-\sqrt5}{1-\sqrt3}\right):\frac{1}{\sqrt7-\sqrt5}\)
\(=\left(-\frac{\sqrt7\left(\sqrt2-1\right)}{\sqrt2-1}-\frac{\sqrt5\left(\sqrt3-1\right)}{\sqrt3-1}\right)\left(\sqrt7-\sqrt5\right)\)
\(=-\left(\sqrt7+\sqrt5\right)\left(\sqrt7-\sqrt5\right)=-\left(7-5\right)=-2\)
