To simplify the expression [tex]\(\left(\frac{1}{4 a b}\right)^{-2}\)[/tex], we can follow these steps:
1. Understand the Negative Exponent Rule:
The rule for negative exponents states that [tex]\(\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n\)[/tex]. In other words, flipping the fraction and changing the sign of the exponent.
Therefore,
[tex]\[
\left(\frac{1}{4 a b}\right)^{-2} = \left(\frac{4 a b}{1}\right)^2.
\][/tex]
2. Simplify the Fraction:
The fraction [tex]\(\frac{4 a b}{1}\)[/tex] simplifies directly to [tex]\(4 a b\)[/tex].
3. Apply the Positive Exponent:
Now we need to raise [tex]\(4 a b\)[/tex] to the power of 2:
[tex]\[
(4 a b)^2.
\][/tex]
4. Apply the Power to Each Term in the Product:
[tex]\[
(4 a b)^2 = 4^2 \cdot a^2 \cdot b^2.
\][/tex]
The exponent applies to each factor within the parentheses.
5. Calculate Each Part:
[tex]\[
4^2 = 16,
\][/tex]
and then
[tex]\[
(4 a b)^2 = 4^2 \cdot a^2 \cdot b^2 = 16 a^2 b^2.
\][/tex]
So, the simplified form of the expression [tex]\(\left(\frac{1}{4 a b}\right)^{-2}\)[/tex] is:
[tex]\[
\boxed{16 a^2 b^2}.
\][/tex]
