The expression in the image asks for the simplified form of:

$\left( \frac{a^3}{2b^{-2}} \right)^{-2}$

Let's simplify it step by step:

Step 1: Rewrite the denominator

The expression in the denominator, $b^{-2}$, can be rewritten as:

$b^{-2} = \frac{1}{b^2}$

So, the fraction becomes:

$\frac{a^3}{2b^{-2}} = \frac{a^3}{2 \times \frac{1}{b^2}} = \frac{a^3 b^2}{2}$

Step 2: Apply the exponent of -2

Now, we need to raise this fraction to the power of $-2$:

$\left( \frac{a^3 b^2}{2} \right)^{-2}$

Using the property of exponents $\left( \frac{A}{B} \right)^{-n} = \frac{B^n}{A^n}$, we get:

$\frac{2^2}{(a^3 b^2)^2} = \frac{4}{a^6 b^4}$

Final Simplified Form:

The simplified form of the expression is:

$\frac{4}{a^6 b^4}$

Would you like further details or explanations?

Here are five related questions you can explore:

1. How do you simplify fractions with negative exponents?
2. What are the rules for multiplying exponents in expressions like $(a^3 b^2)^2$?
3. How do we handle negative exponents in both the numerator and denominator?
4. Can the laws of exponents be applied to complex numbers?
5. What are the applications of exponent rules in real-life problems?

Tip: Always remember that a negative exponent flips the base to the denominator (or numerator, depending on its position), making it positive.