jmc

First, we simplify the radicals as much as possible. We have $\sqrt{54}=\sqrt{2\cdot 3^3}=\sqrt{2\cdot 3\cdot 3^2}=3\sqrt{2\cdot 3}=3\sqrt{6}$, and $\sqrt{32}=\sqrt{2^5}=\sqrt{2^4\cdot 2}=4\sqrt{2}$. Therefore, we have $$\begin{array}{rl}\sqrt{54}\cdot \sqrt{32}\cdot \sqrt{6}& =(3\sqrt{6})(4\sqrt{2})(\sqrt{6})=3\cdot 4\sqrt{6}\cdot \sqrt{2}\sqrt{6}\\ & =12\sqrt{2}(\sqrt{6}\sqrt{6})=(12\sqrt{2})(6)=\boxed{72\sqrt{2}}.\end{array}$$