a)
\frac{1}{3\sqrt2}=\frac{1\sqrt2}{3\sqrt2*\sqrt2}=\frac{\sqrt2}{3*2}=\frac{\sqrt2}{6}
b)
\frac{\sqrt2}{2\sqrt3}=\frac{\sqrt2*\sqrt3}{2\sqrt3*\sqrt3}=\frac{\sqrt6}{2*3}=\frac{\sqrt6}{6}
c)
\frac{1+2\sqrt3}{\sqrt3}=\frac{(1+2\sqrt3)*\sqrt3}{\sqrt3*\sqrt3}=\frac{\sqrt3+2*3}{3}=\frac{\sqrt3+6}{3}
d)
\frac{3}{1+\sqrt2}=\frac{3(1-\sqrt2)}{(1+\sqrt2)(1-\sqrt2)}=\frac{3-3\sqrt2}{1-2}=\frac{3-3\sqrt2}{-1}=-3+3\sqrt2
e)
\frac{\sqrt2}{\sqrt3-3}=\frac{\sqrt2(\sqrt3+3)}{(\sqrt3-3)(\sqrt3+3)}=\frac{\sqrt6+3\sqrt2}{3-9}=\frac{\sqrt6+3\sqrt2}{-6}=
f)
\frac{1-\sqrt2}{1+\sqrt2}=\frac{(1-\sqrt2)(1-\sqrt2)}{(1+\sqrt2)(1-\sqrt2)}=\frac{(1-\sqrt2)^2}{1-2}=\frac{1-2\sqrt2+2}{-1}=\frac{3-2\sqrt2}{-1}=-3+2\sqrt2
g)
\frac{-2+\sqrt7}{-\sqrt7-2}=\frac{2-\sqrt7}{\sqrt7+2}=\frac{(2-\sqrt7)(\sqrt7-2)}{(\sqrt7+2)(\sqrt7-2)}=\frac{2\sqrt7-4-7+2\sqrt7}{7-4}=\frac{4\sqrt7-11}{3}
http://www.wolframalpha.com/input/?i=%28-2%2Bsqrt7%29%2F%28-sqrt7-2%29