To evaluate the expression, we can use the laws of exponents, which state that \(a^{m/n} = (a^m)^{1/n}\) and \(a^{-n} = \frac{1}{a^n}\). Applying these laws, we have:
\(\frac{27^{\frac{1}{3}}}{16^{-\frac{1}{4}}} = \frac{(3^3)^{\frac{1}{3}}}{(2^4)^{-\frac{1}{4}}} = \frac{3^{3\cdot\frac{1}{3}}}{2^{4\cdot(-\frac{1}{4})}} = \frac{3}{2^{-1}} = 3 \cdot 2 = 6\)
Therefore, the answer is 6.