problem: Simplify and rationalize the denominator: $$\frac{8}{3\sqrt{3}+ 7\sqrt{27}}.$$
solution: 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*}