\frac{2}{2\sqrt{5}}=\frac{2\sqrt{5}}{2\sqrt{5}*\sqrt{5}}=\frac{2\sqrt{5}}{2*5}=\frac{\sqrt{5}}{5}
\frac{\sqrt{3}-1}{\sqrt{7}}=\frac{(\sqrt{3}-1)*\sqrt{7}}{\sqrt{7}*\sqrt{7}}=\frac{\sqrt{21}-\sqrt{7}}{7}
\frac{2\sqrt{7}}{2+\sqrt{7}}=\frac{2\sqrt{7}*(2-\sqrt{7})}{(2+\sqrt{7})(2-\sqrt{7})}=\frac{4\sqrt{7}-2*7}{2^2-(\sqrt{7})^2}=\frac{4\sqrt{7}-14}{4-7}=-\frac{4\sqrt{7}-14}{3}
\frac{3+2\sqrt{3}}{2\sqrt{3}-3}=\frac{(3+2\sqrt{3})(2\sqrt+3)}{(2\sqrt{3}-3)(2\sqrt{3}+3)}=\frac{(3+2\sqrt{3})^2}{(2\sqrt{3})^2-3^2}=\frac{3^2+2*3*2\sqrt{3}+(2\sqrt{3})^2}{4*3-9}=\frac{9+12\sqrt{3}+4*3}{3}=\frac{12\sqrt{3}+21}{3}=\frac{3(4\sqrt{3}+7)}{3}=4\sqrt{3}+7