Answer
$$ \frac{4}{37} $$
Work Step by Step
Since
\begin{align*}
0.108108108 &= 0.108+0.000108+0.000000108\\
&=0.108(1)+0.12(10^{-3})+ 0.12(10^{-6})+\cdots \\
&=0.108[1+\frac{1}{10^3}+\frac{1}{10^6}+\cdots ]
\end{align*}
Since
$$1+\frac{1}{10^3}+\frac{1}{10^6}+\cdots $$
which is a geometric series and has the sum $$\frac{1}{1-\frac{1}{1000}}= \frac{1000}{999} $$
Hence
$$0.108108108= 0.108 \frac{1000}{999}= \frac{108}{999} =\frac{4}{37}$$
