Answer
$\sqrt[10]{5^{7}}$
Work Step by Step
Recall the rational exponent property (pg. 382):
$\sqrt[n]{a}=a^{\frac{1}{n}}$
Applying this property, we get:
$\left(\sqrt[3]5\right)\left(\sqrt[5]5\right)=\left(5^{\frac{1}{2}}\right)\left(5^{\frac{1}{5}}\right)$
Next, recall the basic exponent property (pg. 360):
$a^ma^n=a^{m+n}$
Applying this property to our last equation, we get:
\begin{align*}
\left(5^{\frac{1}{2}}\right)\left(5^{\frac{1}{5}}\right)&=5^{\frac{1}{2}+\frac{1}{5}}\\
&=5^{\frac{5}{10}+\frac{2}{10}}\\
&=5^{\frac{5+2}{10}}\\
&=5^{\frac{7}{10}}
\end{align*}
Now, recall the rational exponent property (pg. 382):
$a^{\frac{m}{n}}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m$
We use this property to rewrite the result as a radical:
$5^{\frac{7}{10}}=\sqrt[10]{5^{7}}$
