Simplify $(576)^\frac{1}{4}(216)^\frac{1}{2}$.
Simplify the two factors first.  Factor $2^4$ from $576$ to get $36$, and since $36=6^2$, the fourth root of $576$ is $(2^\frac44)(6^\frac24)=2\sqrt{6}$.  Because $6^2$ evenly divides $216$, the square root of $216$ is $(6^\frac22)(6^\frac12)=6\sqrt{6}$.  Multiply those two quantities together to obtain $\boxed{72}$ as an answer.