$a)\frac{3}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{3\sqrt{2}}{\sqrt{2}\cdot \sqrt{2}}=\frac{3\sqrt{2}}{\sqrt{4}}=\frac{3\sqrt{2}}{2}$  

$b)\frac{15}{\sqrt{5}}=\frac{15}{\sqrt{5}}\cdot \frac{\sqrt{5}}{\sqrt{5}}=\frac{15\sqrt{5}}{\sqrt{25}}=\frac{15\sqrt{5}}{5}=3\sqrt{5}$ $b)\frac{15}{\sqrt{5}}=\frac{15}{\sqrt{5}}\cdot \frac{\sqrt{5}}{\sqrt{5}}=\frac{15\sqrt{5}}{\sqrt{25}}=\frac{15\sqrt{5}}{5}=3\sqrt{5}$  

$c)\frac{2\sqrt{3}}{3\sqrt{6}}=\frac{2\sqrt{3}}{3\sqrt{6}}\cdot \frac{\sqrt{6}}{\sqrt{6}}=\frac{2\sqrt{3}\cdot \sqrt{6}}{3\sqrt{6}\cdot \sqrt{6}}=\frac{2\sqrt{18}}{3\cdot 6}=\frac{2\sqrt{9\cdot 2}}{18}=\frac{2\sqrt{9}\cdot \sqrt{2}}{18}=\frac{2\cdot 3\cdot \sqrt{2}}{18}=\frac{6\sqrt{2}}{18}=\frac{\sqrt{2}}{3}$   

$d)\frac{\sqrt{7}}{\sqrt{70}}=\frac{\sqrt{7}}{\sqrt{70}}\cdot \frac{\sqrt{70}}{\sqrt{70}}=\frac{\sqrt{7\cdot 70}}{\sqrt{{70}^2}}=\frac{\sqrt{490}}{70}=\frac{\sqrt{49\cdot 10}}{70}=\frac{\sqrt{49}\cdot \sqrt{10}}{70}=\frac{7\sqrt{10}}{70}=\frac{\sqrt{10}}{10}$  

$e)\frac{1}{\sqrt[3]{100}}=\frac{1}{\sqrt[3]{100}}\cdot \frac{\sqrt[3]{{100}^2}}{\sqrt[3]{{100}^2}}=\frac{\sqrt[3]{{100}^2}}{\sqrt[3]{{100}^3}}=\frac{\sqrt[3]{10000}}{100}=\frac{\sqrt[3]{1000\cdot 10}}{100}=\frac{\sqrt[3]{1000}\cdot \sqrt[3]{10}}{100}=\frac{10\cdot \sqrt[3]{10}}{100}=\frac{\sqrt[3]{10}}{10}$   

$f)\frac{\sqrt[3]{3}}{\sqrt[3]{24}}=\sqrt[3]{\frac{3}{24}}=\sqrt[3]{\frac{1}{8}}=\frac{1}{2}$