To rationalize a given number:
[tex]\sqrt[3]{\frac{3a^7}{2b}}[/tex][tex]\mathrm{Apply\: radical\: rule}\colon\quad \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}[/tex][tex]\begin{gathered} \frac{\sqrt[3]{3a^7}}{\sqrt[3]{2b^7}}=\frac{\sqrt[3]{3a^7}}{\sqrt[3]{2b^7}}\frac{(2b)^{\frac{2}{3}}}{(2b)^{\frac{2}{3}}} \\ \text{ multiply by }the\text{ conjugate } \\ simplify\text{ }\sqrt[3]{3a^7}(2b)^{\frac{2}{3}}\colon\sqrt[3]{3}.2^{\frac{2}{3}}a^2\sqrt[3]{a}b^{\frac{2}{3}} \\ =\sqrt[3]{2b^{}}(2b)^{\frac{2}{3\text{ }}}=2b \\ \frac{\sqrt[3]{3}\cdot\:2^{\frac{2}{3}}a^2\sqrt[3]{a}b^{\frac{2}{3}}}{2b} \end{gathered}[/tex]
Hence the correct answer is
[tex]\frac{\sqrt[3]{3}\cdot\: 2^{\frac{2}{3}}a^2\sqrt[3]{a}b^{\frac{2}{3}}}{2b}[/tex]
