Variance of exp(X) for a gamma variable

Let $X$ follow a gamma distribution with shape $8$ and scale $\tfrac13$ (equivalently rate $3$), written $\text{Gamma}(8,\tfrac13)$. The variance of $e^{X}$ can be expressed as $a^{b}-\left(\tfrac{a}{2}\right)^{c}$ with integers $a,b,c$. Find $abc$.

$384$. Use the moment generating function $M(t)=(1-t/3)^{-8}$: $\operatorname{Var}(e^X)=M(2)-M(1)^2=3^8-(3/2)^{16}$. So $a=3$, $b=8$, $c=16$ and $abc=384$.