a: \(2\sqrt{48}+6\sqrt{\frac13}-4\sqrt{12}\)
\(=2\cdot4\sqrt3+2\sqrt3-4\cdot2\sqrt3\)
\(=8\sqrt3+2\sqrt3-8\sqrt3=2\sqrt3\)
b: \(\sqrt{\left(\sqrt2-\sqrt3\right)^2}\cdot\sqrt{\frac{2}{5-2\sqrt6}}\)
\(=\sqrt{\frac{2}{5-2\sqrt6}\cdot\left(5-2\sqrt6\right)}=\sqrt2\)
c: \(\frac{2}{\sqrt5+1}+\sqrt{\frac{2}{3-\sqrt5}}-\frac{5}{\sqrt5}\)
\(=\frac{2\left(\sqrt5-1\right)}{\left(\sqrt5+1\right)\left(\sqrt5-1\right)}+\sqrt{\frac{2\left(3+\sqrt5\right)}{\left(3-\sqrt5\right)\left(3+\sqrt5\right)}}-\sqrt5\)
\(=\frac{2\left(\sqrt5-1\right)}{5-1}+\sqrt{\frac{6+2\sqrt5}{4}}-\sqrt5=\frac{\sqrt5-1}{2}-\frac{2\sqrt5}{2}+\frac{\sqrt5+1}{2}=\frac{-\sqrt5-1+\sqrt5+1}{2}=0\)
d: \(\frac{\sqrt6-\sqrt3}{\sqrt2-1}+\sqrt{\frac{2}{4+\sqrt{15}}}\)
\(=\frac{\sqrt3\left(\sqrt2-1\right)}{\sqrt2-1}+\sqrt{\frac{2\left(4-\sqrt{15}\right)}{\left(4+\sqrt{15}\right)\left(4-\sqrt{15}\right)}}\)
\(=\sqrt3+\sqrt{2\left(4-\sqrt{15}\right)}=\sqrt3+\sqrt{8-2\sqrt{15}}\)
\(=\sqrt3+\sqrt{\left(\sqrt5-\sqrt3\right)^2}=\sqrt3+\sqrt5-\sqrt3=\sqrt5\)
