Answer
$248.25$ years
Work Step by Step
Given \begin{equation}
a=\sqrt[3]{P^2}= P^{\frac{2}{3}}.
\end{equation} First solve for $P$ in term of $a$. \begin{equation}
\begin{aligned}
P^{\frac{2}{3}}&=a\\
\left(P^{\frac{2}{3}} \right)^{\frac{3}{2}}& =\left( a\right)^{\frac{3}{2}}\\
P& = a^{\frac{3}{2}}.
\end{aligned}
\end{equation} Now, set $a= 39.5$ to find the number of years that it takes Pluto to orbit the sun.
\begin{equation}
\begin{aligned}
P& = 39.5^{\frac{3}{2}}\\
&\approx 248.25.
\end{aligned}
\end{equation} It will take Pluto about $248.25$ years to orbit the sun.
