Simplify and rationalize the denominator: $$\frac{8}{3\sqrt{3}+ 7\sqrt{27}}.$$
We begin by recognizing that since $27 = 3^3$, $7\sqrt{27}$ simplifies to $7\cdot3\sqrt{3}=21\sqrt{3}$. Our expression then becomes  \begin{align*}
\frac{8}{3\sqrt{3}+21\sqrt{3}} & = \frac{8}{24\sqrt{3}} \\
& = \frac{1}{3\sqrt{3}} \\
& = \boxed{\frac{\sqrt{3}}{9}}.
\end{align*}