Answer
$\sqrt[30]{{{t}^{7}}}$
Work Step by Step
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 $\sqrt[30]{{{t}^{7}}}$.
