The problem asks for the simplified form of the expression:

$\left(\frac{x^7 y^{-3}}{x^{-4} y^6}\right)^{-2}$

Let's break it down step by step:

Step 1: Simplify the expression inside the parentheses.

We will simplify the terms in the numerator and denominator separately:

• The numerator is $x^7 y^{-3}$
• The denominator is $x^{-4} y^6$

Using the property of exponents for division $\frac{a^m}{a^n} = a^{m-n}$, we can simplify:

$\frac{x^7}{x^{-4}} = x^{7 - (-4)} = x^{7+4} = x^{11}$

$\frac{y^{-3}}{y^6} = y^{-3 - 6} = y^{-9}$

Thus, the simplified expression inside the parentheses is:

$\frac{x^{11}}{y^9}$

Step 2: Apply the exponent $-2$ to the entire expression.

Now, apply the exponent $-2$ to both $x^{11}$ and $y^{-9}$:

$\left( \frac{x^{11}}{y^9} \right)^{-2} = \frac{y^{9 \times 2}}{x^{11 \times 2}} = \frac{y^{18}}{x^{22}}$

Final Answer:

The simplified form of the given expression is:

$\frac{y^{18}}{x^{22}}$

So, the correct answer is option b.

Would you like a more detailed explanation or have any questions about this?

Here are 5 related questions to explore this concept further:

1. How do exponent rules apply when multiplying or dividing terms with the same base?
2. What happens when an expression is raised to a negative exponent?
3. Can you simplify expressions with different variables using the same rules?
4. How do fractional exponents relate to roots (like square roots or cube roots)?
5. What strategies can you use to simplify complex expressions involving multiple variables?

Tip: When simplifying expressions, always start by applying exponent rules to both the numerator and denominator, and then handle any remaining operations such as raising to an exponent.