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?

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.