Let's simplify the expression:

$\left(4a^2b^{-2}\right)^{-2}$

Step 1: Distribute the exponent $-2$ to each factor inside the parentheses.

Apply the exponent $-2$ to each part of the expression:

$\left(4\right)^{-2} \cdot \left(a^2\right)^{-2} \cdot \left(b^{-2}\right)^{-2}$

Step 2: Simplify each term.

• $\left(4\right)^{-2} = \frac{1}{4^2} = \frac{1}{16}$
• $\left(a^2\right)^{-2} = a^{2 \times (-2)} = a^{-4} = \frac{1}{a^4}$
• $\left(b^{-2}\right)^{-2} = b^{-2 \times (-2)} = b^{4}$

Step 3: Combine the terms.

Now, multiply the simplified terms together:

$\frac{1}{16} \cdot \frac{1}{a^4} \cdot b^4 = \frac{b^4}{16a^4}$

Final Answer:

$\boxed{\frac{b^4}{16a^4}}$

Would you like more details or have any questions?

Here are some related questions you might find helpful:

1. How do exponents distribute over products?
2. How do you handle negative exponents in algebraic expressions?
3. What are the steps to simplify complex exponential expressions?
4. Can you demonstrate simplifying a similar expression with fractions?
5. How do you simplify expressions with multiple variables and exponents?

Tip: Always simplify the expression inside the parentheses first before applying the outer exponent.