jmc

We begin by recognizing that since $27=3^3$, $7\sqrt{27}$ simplifies to $7\cdot 3\sqrt{3}=21\sqrt{3}$. Our expression then becomes $$\begin{array}{rl}\frac{8}{3\sqrt{3}+21\sqrt{3}}& =\frac{8}{24\sqrt{3}}\\ & =\frac{1}{3\sqrt{3}}\\ & =\boxed{\frac{\sqrt{3}}{9}}.\end{array}$$