To simplify the expression [tex]\(\left(\frac{1}{4ab}\right)^{-2}\)[/tex], let's go through it step by step.
1. Understanding Negative Exponents:
The expression [tex]\(\left(\frac{1}{4ab}\right)^{-2}\)[/tex] involves a negative exponent. Recall that:
[tex]\[
(x^{-n}) = \frac{1}{x^n}
\][/tex]
and
[tex]\[
\left(\frac{1}{x}\right)^{-n} = x^n
\][/tex]
Therefore,
[tex]\[
\left(\frac{1}{4ab}\right)^{-2} = (4ab)^2
\][/tex]
2. Applying the Exponent:
Now, we need to square the term inside the parentheses:
[tex]\[
(4ab)^2
\][/tex]
3. Exponent Distributes Over Multiplication:
When raising a product to an exponent, apply the exponent to each factor inside:
[tex]\[
(4ab)^2 = 4^2 \cdot (a)^2 \cdot (b)^2
\][/tex]
4. Calculating Each Part:
[tex]\[
4^2 = 16, \quad a^2 = a^2, \quad b^2 = b^2
\][/tex]
5. Combining the Results:
[tex]\[
16 \cdot a^2 \cdot b^2 = 16a^2b^2
\][/tex]
Thus, the simplified expression is:
[tex]\[
16a^2b^2
\][/tex]
The correct answer is:
[tex]\[
\boxed{16a^2b^2}
\][/tex]
