Question

Rationalize the numerator. Assume that all expressions under radicals represent positive numbers.3/3a2b

Answer 1

To rationalize a given number:

`\sqrt[3]{(3a^7)/(2b)}`
`\mathrm{Apply\: radical\: rule}\colon\quad \sqrt[n]{(a)/(b)}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}`
`\begin{gathered} \frac{\sqrt[3]{3a^7}}{\sqrt[3]{2b^7}}=\frac{\sqrt[3]{3a^7}}{\sqrt[3]{2b^7}}\frac{(2b)^{(2)/(3)}}{(2b)^{(2)/(3)}} \\ \text{ multiply by }the\text{ conjugate } \\ simplify\text{ }\sqrt[3]{3a^7}(2b)^{(2)/(3)}\colon\sqrt[3]{3}.2^{(2)/(3)}a^2\sqrt[3]{a}b^{(2)/(3)} \\ =\sqrt[3]{2b^{}}(2b)^{\frac{2}{3\text{ }}}=2b \\ \frac{\sqrt[3]{3}\cdot\:2^{(2)/(3)}a^2\sqrt[3]{a}b^{(2)/(3)}}{2b} \end{gathered}`

Hence the correct answer is

`\frac{\sqrt[3]{3}\cdot\: 2^{(2)/(3)}a^2\sqrt[3]{a}b^{(2)/(3)}}{2b}`