We are given the expression:

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

Let's simplify it step by step.

Step 1: Simplify the fraction inside the parentheses

Start by simplifying the fraction $\frac{x^4 y^{-3}}{x^{-4} y^3}$.

Using the property of exponents $\frac{a^m}{a^n} = a^{m-n}$, we simplify the powers of $x$ and $y$ separately:

1. $\frac{x^4}{x^{-4}} = x^{4 - (-4)} = x^{4 + 4} = x^8$
2. $\frac{y^{-3}}{y^3} = y^{-3 - 3} = y^{-6}$

Thus, the expression inside the parentheses simplifies to:

$x^8 y^{-6}$

Step 2: Apply the exponent outside the parentheses

Now, raise the simplified expression to the power of $-2$. Using the rule $(a^m)^n = a^{m \cdot n}$, we get:

$\left( x^8 y^{-6} \right)^{-2} = x^{8 \cdot (-2)} y^{-6 \cdot (-2)} = x^{-16} y^{12}$

Final Answer:

The simplified expression is:

$x^{-16} y^{12}$

Let me know if you need more details or have any questions about this solution!

Related Questions:

1. How would you simplify an expression like $\left( \frac{a^m b^n}{a^p b^q} \right)^{-r}$?
2. What happens if you have a negative exponent inside a fraction like $\frac{x^{-3}}{y^2}$?
3. How do you simplify a complex rational expression with multiple variables and exponents?
4. Can you explain the distributive property of exponents in more detail?
5. How would the result change if the original expression were raised to the power of $3$ instead of $-2$?

Tip:

When working with exponents, always simplify each part of the fraction individually first before applying the outer exponent. This often reduces the complexity of the expression.