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The expression to simplify is:

\[
\left( \frac{x^3 y^{-3}}{x^2 y} \right)^{-3}
\]

Let's break down the simplification step by step:

### Step 1: Simplify the expression inside the parentheses
We can simplify the fraction inside the parentheses by applying the laws of exponents:

\[
\frac{x^3 y^{-3}}{x^2 y} = x^{3-2} y^{-3-1} = x^1 y^{-4}
\]

This simplifies to:

\[
x y^{-4}
\]

### Step 2: Apply the exponent of \(-3\)
Now, apply the exponent \(-3\) to the simplified expression \(x y^{-4}\):

\[
(x y^{-4})^{-3} = x^{-3} y^{12}
\]

### Final Simplified Expression:
\[
x^{-3} y^{12}
\]

Would you like more details on any of the steps or have any further questions?

Here are five related questions for practice:
1. How do you simplify expressions with negative exponents?
2. What happens when you raise a product of terms to a power?
3. Can you simplify \(\left( \frac{x^2}{y^{-1}} \right)^{-2}\)?
4. How do you divide powers of the same base with different exponents?
5. What is the difference between applying exponents to a single term versus multiple terms?

**Tip:** When simplifying expressions, always handle negative exponents carefully by converting them into positive exponents whenever possible!

Learn how to simplify the expression (x^3 y^{-3} / x^2 y)^{-3} step by step using the laws of exponents. This problem covers key concepts like simplifying fractions with exponents and handling negative powers. Suitable for students in Grades 9-11.

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