a: \(\frac32\cdot\sqrt6+2\cdot\sqrt{\frac23}-4\cdot\sqrt{\frac32}-\frac{\sqrt{24}}{12}\)
\(=\frac32\cdot\sqrt6+2\cdot\sqrt{\frac69}-4\cdot\sqrt{\frac64}-\frac{2\sqrt6}{12}\)
\(=\frac32\sqrt6+\frac23\sqrt6-2\sqrt6-\frac16\sqrt6=\sqrt6\left(\frac32+\frac23-2-\frac16\right)\)
\(=\sqrt6\left(\frac96+\frac46-\frac{12}{6}-\frac16\right)=0\)
b: \(\frac{3}{\sqrt5+\sqrt2}+\frac{1}{\sqrt2-1}-\frac{4}{3-\sqrt5}\)
\(=\frac{3\left(\sqrt5-\sqrt2\right)}{\left(\sqrt5+\sqrt2\right)\left(\sqrt5-\sqrt2\right)}+\frac{\left(\sqrt2+1\right)}{\left(\sqrt2-1\right)\left(\sqrt2+1\right)}-\frac{4\left(3+\sqrt5\right)}{\left(3-\sqrt5\right)\left(3+\sqrt5\right)}\)
\(=\sqrt5-\sqrt2+\sqrt2+1-\left(3+\sqrt5\right)=\sqrt5+1-3-\sqrt5\)
=1-3
=-2
c: \(\left(\frac{\sqrt{14}-\sqrt7}{1-\sqrt2}+\frac{\sqrt{15}-\sqrt5}{1-\sqrt3}\right):\frac{1}{\sqrt7-\sqrt5}\)
\(=\left(-\frac{\sqrt7\left(\sqrt2-1\right)}{\sqrt2-1}-\frac{\sqrt5\left(\sqrt3-1\right)}{\sqrt3-1}\right)\left(\sqrt7-\sqrt5\right)\)
\(=\left(-\sqrt7-\sqrt5\right)\left(\sqrt7-\sqrt5\right)=-\left(\sqrt7+\sqrt5\right)\left(\sqrt7-\sqrt5\right)=-\left(7-5\right)=-2\)
d: \(\sqrt{5-\sqrt{22+2\sqrt2}}\cdot\sqrt{5+\sqrt{22+2\sqrt2}}\)
\(=\sqrt{5^2-\left(22+2\sqrt2\right)}\)
\(=\sqrt{25-22-2\sqrt2}=\sqrt{3-2\sqrt2}=\sqrt{\left(\sqrt2-1\right)^2}=\sqrt2-1\)
