## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$\sqrt[30]{{{t}^{7}}}$
Formula to be used: Quotient rule for radicals: For any real number $\left( \sqrt[n]{a}\text{ and }\sqrt[n]{b} \right)$, $b\ne 0$. $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$ Consider expression $\frac{\sqrt[3]{t}}{\sqrt[10]{t}}$, Convert to the exponential notation, $\frac{\sqrt[3]{t}}{\sqrt[10]{t}}=\frac{{{t}^{\frac{1}{3}}}}{{{t}^{\frac{1}{10}}}}$ Add exponent, \begin{align} & \frac{{{t}^{\frac{1}{3}}}}{{{t}^{\frac{1}{10}}}}={{t}^{\left( \frac{1}{3}-\frac{1}{10} \right)}} \\ & ={{t}^{\left( \frac{10-3}{30} \right)}} \\ & ={{t}^{\left( \frac{7}{30} \right)}} \end{align} Convert back to radical notation, $\sqrt[30]{{{t}^{7}}}$ Thus, the expression $\frac{\sqrt[3]{t}}{\sqrt[10]{t}}$ can be simplified as $\sqrt[30]{{{t}^{7}}}$.