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The given expression is:

\[
\left( \frac{6a^2}{3b^{-2}} \right)^{-2}
\]

Let's evaluate it step by step:

### Step 1: Simplify the fraction inside the parentheses
\[
\frac{6a^2}{3b^{-2}} = 2a^2b^2
\]
This is because:
- \( \frac{6}{3} = 2 \)
- \( b^{-2} \) in the denominator becomes \( b^2 \) when moved to the numerator.

So now we have:
\[
\left( 2a^2b^2 \right)^{-2}
\]

### Step 2: Apply the negative exponent
When raising a term to a negative exponent, we take the reciprocal and raise the term to the positive exponent:
\[
\left( 2a^2b^2 \right)^{-2} = \frac{1}{(2a^2b^2)^2}
\]

### Step 3: Square the terms inside the parentheses
\[
(2a^2b^2)^2 = 2^2 \cdot (a^2)^2 \cdot (b^2)^2 = 4a^4b^4
\]

So, we have:
\[
\frac{1}{4a^4b^4}
\]

### Final Answer:
\[
\boxed{\frac{1}{4a^4b^4}}
\]

Would you like more details or have any questions?

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Here are 5 related questions for further practice:
1. How do you simplify expressions with negative exponents?
2. What happens when you raise a power to another power?
3. How would you simplify \( (3x^{-1}y^2)^{-3} \)?
4. How do negative exponents affect fractions?
5. What is the general rule for distributing exponents over multiplication?

**Tip:** When dealing with negative exponents, always remember that they indicate taking the reciprocal of the base.

Step-by-step solution for simplifying and evaluating the expression (6a^2 / 3b^-2)^-2. This problem covers algebraic concepts, focusing on the rules of exponents and simplification techniques. Suitable for students in Grades 8-10.

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