a)

\(\frac{1}{81}^{-\frac{1}{4}}\)

\(=(\frac{1}{3})^{-\frac{1}{4}}\)

\(=(\frac{1}{3})\) (product of power property)

\((\frac{1}{3})^{-1}=3\)

b)

\(x^2 y^-4\)

\(=x^{-2}\cdot y^{-4}\)

\(\frac{1}{x^{2}}\cdot\frac{1}{y^{4}}\) (negative power property)

c)

\((2x)^{-2}(16x^{2y})^{\frac{1}{2}}\)

\(=2^{-2}\cdot x^{-2}\cdot2^{\frac{4}{2}}\cdot x^{\frac{2}{2}}\cdot y^{\frac{1}{2}}\) (product of power property)

\(=2^{(2-2)}\cdot x^{(1-2)}\cdot y^{\frac{1}{2}}\)

\(=1\cdot x^{-1}\cdot y(\frac{1}{2})\)