The expression in the image appears to be:

$\left( \frac{4x^{-4}}{y^2} \right)^{-1} \left( \frac{3y^{-2}}{x^4} \right)^2$

To simplify this, follow these steps:

Step 1: Apply the negative exponents

Using the rule $a^{-n} = \frac{1}{a^n}$, we rewrite the expression:

$\left( \frac{4x^{-4}}{y^2} \right)^{-1} = \frac{y^2}{4x^{-4}} = \frac{y^2}{4} \cdot x^4$ and $\left( \frac{3y^{-2}}{x^4} \right)^2 = \frac{9y^{-4}}{x^8} = \frac{9}{y^4 x^8}$

Step 2: Multiply the results

Now, multiply the two simplified expressions:

$\frac{y^2}{4} \cdot x^4 \cdot \frac{9}{y^4 x^8}$

This simplifies to:

$\frac{9x^4y^2}{4y^4 x^8} = \frac{9}{4x^4 y^2}$

So, the simplified expression is:

$\frac{9}{4x^4 y^2}$

Would you like further details or have any questions about this process?

Here are some related questions to explore:

1. What does the rule for negative exponents state, and how do we apply it?
2. How do you handle powers when multiplying terms with the same base?
3. What happens when you raise a fraction to a power?
4. What is the rule for dividing terms with exponents?
5. How do you simplify expressions with both positive and negative exponents?

Tip: Always convert negative exponents to fractions first—it's often easier to simplify the expression.