Algebra

First, let's isolate a^3 as much as possible. 

We have

$\frac{1}{64a^3+7}=7\\448a^{3}+49=1\\ 448a^3=-48\\ a^3=-48/448$

From this, we get that

$a=\sqrt[3]{-28/3}$

This is actually the only real factor for a, so it is our final answer. 

Thanks! :)