$\frac{4}{\sqrt{2}}=\frac{4\ast <!--\; \ast \; -->\sqrt{2}}{\sqrt{2}\ast <!--\; \ast \; -->\sqrt{2}}=\frac{4\sqrt{2}}{(\sqrt{2}{)}^2}=\frac{4\sqrt{2}}{2}=2\sqrt{2}$

$\frac{12}{\sqrt{6}}=\frac{12\ast <!--\; \ast \; -->\sqrt{6}}{\sqrt{6}\ast <!--\; \ast \; -->\sqrt{6}}=\frac{12\sqrt{6}}{(\sqrt{6}{)}^2}=\frac{12\sqrt{6}}{6}=2\sqrt{6}$

$\frac{18}{\sqrt{5}}=\frac{18\ast <!--\; \ast \; -->\sqrt{5}}{\sqrt{5}\ast <!--\; \ast \; -->\sqrt{5}}=\frac{18\sqrt{5}}{(\sqrt{5}{)}^2}=\frac{18\sqrt{5}}{5}$

$\frac{m}{\sqrt{n}}=\frac{m\ast <!--\; \ast \; -->\sqrt{n}}{\sqrt{n}\ast <!--\; \ast \; -->\sqrt{n}}=\frac{m\sqrt{n}}{(\sqrt{n}{)}^2}=\frac{m\sqrt{n}}{n}$

$\frac{a}{b\sqrt{b}}=\frac{a\ast <!--\; \ast \; -->\sqrt{b}}{b\sqrt{b}\ast <!--\; \ast \; -->\sqrt{b}}=\frac{a\ast <!--\; \ast \; -->\sqrt{b}}{b(\sqrt{b}{)}^2}=\frac{a\ast <!--\; \ast \; -->\sqrt{b}}{b\ast <!--\; \ast \; -->b}=\frac{a\sqrt{b}}{b^2}$

$\frac{5}{\sqrt{15}}=\frac{5\ast <!--\; \ast \; -->\sqrt{15}}{\sqrt{15}\ast <!--\; \ast \; -->\sqrt{15}}=\frac{5\sqrt{15}}{(\sqrt{15}{)}^2}=\frac{5\sqrt{15}}{15}=\frac{\sqrt{15}}{3}$

$\frac{7}{\sqrt{7}}=\frac{7\ast <!--\; \ast \; -->\sqrt{7}}{\sqrt{7}\ast <!--\; \ast \; -->\sqrt{7}}=\frac{7\sqrt{7}}{(\sqrt{7}{)}^2}=\frac{7\sqrt{7}}{7}=\sqrt{7}$

$\frac{24}{5\sqrt{3}}=\frac{24\ast <!--\; \ast \; -->\sqrt{3}}{5\sqrt{3}\ast <!--\; \ast \; -->\sqrt{3}}=\frac{24\sqrt{3}}{5(\sqrt{3}{)}^2}=\frac{24\sqrt{3}}{5\ast <!--\; \ast \; -->3}=\frac{8\sqrt{3}}{5}$