Simplify the expression \( \left(\frac{1}{4 a b}\right)^{-2} \). Assume \( a \neq 0, b \neq 0 \)

To simplify the expression \( \left(\frac{1}{4 a b}\right)^{-2} \), we can apply the property of exponents that states \( x^{-n} = \frac{1}{x^n} \). Starting with our expression: \[ \left(\frac{1}{4 a b}\right)^{-2} = \frac{1}{\left(\frac{1}{4 a b}\right)^{2}} \] Now, rewriting \( \left(\frac{1}{4 a b}\right)^{2} \): \[ \left(\frac{1}{4 a b}\right)^{2} = \frac{1^2}{(4 a b)^{2}} = \frac{1}{16 a^2 b^2} \] Thus, substituting this back, we get: \[ \frac{1}{\left(\frac{1}{4 a b}\right)^{2}} = \frac{1}{\frac{1}{16 a^2 b^2}} = 16 a^2 b^2 \] Therefore, the simplified expression is: \[ \boxed{16 a^2 b^2} \]