Let's break down the expression step by step and simplify it:

The given expression is:

$\frac{X^{-2} Y^3}{X^3 Y^{-2}} \times X Y$

Step 1: Simplifying the first fraction

We simplify the powers of $X$ and $Y$ separately:

$\frac{X^{-2}}{X^3} = X^{-2 - 3} = X^{-5}$ $\frac{Y^3}{Y^{-2}} = Y^{3 - (-2)} = Y^{3 + 2} = Y^5$

So the simplified fraction becomes:

$\frac{X^{-2} Y^3}{X^3 Y^{-2}} = X^{-5} Y^5$

Step 2: Multiplying by $X Y$

Now, multiply the result by $X Y$:

$(X^{-5} Y^5) \times (X Y) = X^{-5 + 1} Y^{5 + 1} = X^{-4} Y^6$

Final result:

The simplified expression is:

$X^{-4} Y^6$

Let me know if you'd like further clarification or additional details.

---

Here are 5 related questions to further your understanding:

1. How do you simplify expressions with negative exponents?
2. What are the rules for adding and subtracting exponents when multiplying or dividing terms?
3. How would the expression change if there were more variables involved?
4. Can you express the result $X^{-4} Y^6$ without using negative exponents?
5. What happens if the powers of $X$ and $Y$ are fractional instead of integer?

Tip: When multiplying terms with the same base, always add the exponents. When dividing, subtract the exponents.