7: \(\frac{2\sqrt3+3\sqrt2}{\sqrt2+\sqrt3}+\frac12\left(\sqrt2-\sqrt3\right)^2\)
\(=\frac{\sqrt6\left(\sqrt2+\sqrt3\right)}{\sqrt2+\sqrt3}+\frac12\left(5-2\sqrt6\right)\)
\(=\sqrt6+\frac52-\sqrt6=\frac52\)
8: \(\frac{3+2\sqrt3}{\sqrt3}+\frac{2+\sqrt2}{\sqrt2+1}-\left(2+\sqrt3\right)\)
\(=2+\sqrt3+\frac{\sqrt2\left(\sqrt2+1\right)}{\sqrt2+1}-2-\sqrt3\)
\(=\sqrt2\)
9: \(\frac{1}{\sqrt5-2}+\frac{1}{\sqrt5+2}\)
\(=\frac{\sqrt5+2+\sqrt5-2}{\left(\sqrt5-2\right)\left(\sqrt5+2\right)}=\frac{2\sqrt5}{5-4}=2\sqrt5\)
10: \(\frac{2}{4-3\sqrt2}-\frac{2}{4+3\sqrt2}\)
\(=\frac{2\left(4+3\sqrt2\right)-2\left(4-3\sqrt2\right)}{\left(4+3\sqrt2\right)\left(4-3\sqrt2\right)}\)
\(=\frac{8+6\sqrt2-8+6\sqrt2}{16-18}=\frac{12\sqrt2}{-2}=-6\sqrt2\)
11: \(\frac{2+\sqrt2}{1+\sqrt2}-\frac{2-\sqrt2}{1-\sqrt2}\)
\(=\frac{\sqrt2\left(1+\sqrt2\right)}{1+\sqrt2}+\frac{\sqrt2\left(\sqrt2-1\right)}{\sqrt2-1}\)
\(=\sqrt2+\sqrt2=2\sqrt2\)
12: \(\left(\sqrt2-\sqrt6\right)\cdot\sqrt{2+\sqrt3}\)
\(=\left(1-\sqrt3\right)\cdot\sqrt{4+2\sqrt3}\)
\(=\left(1-\sqrt3\right)\left(\sqrt3+1\right)=\left(1-\sqrt3\right)\left(1+\sqrt3\right)\)
=1-3
=-2
