a: \(\sqrt3\cdot\sqrt{12}-\sqrt2\cdot\sqrt{2-\sqrt3}-2\sqrt3\)
\(=\sqrt{36}-\sqrt{4-2\sqrt3}-2\sqrt3\)
\(=6-\sqrt{\left(\sqrt3-1\right)^2}-2\sqrt3=6-2\sqrt3-\sqrt3+1=7-3\sqrt3\)
b: \(\frac{2\sqrt{15}-2\sqrt{10}+\sqrt6-3}{2\sqrt5-2\sqrt{10}-\sqrt3+\sqrt6}\)
\(=\frac{2\sqrt5\left(\sqrt3-\sqrt2\right)-\sqrt3\left(\sqrt3-\sqrt2\right)}{2\sqrt5\left(1-\sqrt2\right)-\sqrt3\left(1-\sqrt2\right)}\)
\(=\frac{\left(\sqrt3-\sqrt2\right)\left(2\sqrt5-\sqrt3\right)}{\left(1-\sqrt2\right)\left(2\sqrt5-\sqrt3\right)}=\frac{\sqrt3-\sqrt2}{1-\sqrt2}=\frac{\left(\sqrt3-\sqrt2\right)\left(1+\sqrt2\right)}{\left.\left(1-\sqrt2\right)\left(1+\sqrt2\right)\right.}\)
\(=\left(\sqrt2-\sqrt3\right)\left(\sqrt2+1\right)=2+\sqrt2-\sqrt6-\sqrt3\)
c: \(\frac{\sqrt2+\sqrt3+\sqrt6+\sqrt8+16}{\sqrt2+\sqrt3+\sqrt4}\)
\(=\frac{\sqrt2+\sqrt3+\sqrt4+\sqrt4+\sqrt6+\sqrt8}{\sqrt2+\sqrt3+\sqrt4}\)
\(=\frac{\left(\sqrt2+\sqrt3+\sqrt4\right)\left(1+\sqrt2\right)}{\sqrt2+\sqrt3+\sqrt4}=1+\sqrt2\)
