Answer
$\displaystyle\color{blue}{\left(\frac{2+e^3}{3}\right)^{10} \;\approx\; 467,\!591,\!999.417}$
Work Step by Step
$\begin{align*}
E(e^{3X}) &= E(e^{Xt})\Biggr\vert_{t=3} \\
&= M_X(t)\Biggr\vert_{t=3} & \text{[ Def. of mgf ]}\\
&= M_X(3) \\
&= \left(\frac{2}{3} + \frac{1}{3}e^3\right)^{10} & \text{[ since }\ X\sim \text{Binomial}(10,{\scriptsize\frac{1}{3}}), \\
& & \text{see Example 3.12.2, p. 208 ]} \\
\color{blue}{E(e^{3X})}\ &\color{blue}{= \left(\frac{2+e^3}{3}\right)^{10} \;\approx\; 467,591,999.417}
\end{align*}$
