You are asking to simplify the expression \( \frac{X^2 Y^3}{X^3 Y^{-2}} \times X Y \).

Let’s solve this step by step:

1. First, simplify \( \frac{X^2 Y^3}{X^3 Y^{-2}} \):
   \[
   \frac{X^2}{X^3} = X^{2-3} = X^{-1}
   \]
   \[
   \frac{Y^3}{Y^{-2}} = Y^{3 - (-2)} = Y^{3+2} = Y^5
   \]
   So, \( \frac{X^2 Y^3}{X^3 Y^{-2}} = X^{-1} Y^5 \).

2. Now, multiply by \( X Y \):
   \[
   (X^{-1} Y^5) \times (X Y) = X^{-1+1} Y^{5+1} = X^0 Y^6 = Y^6
   \]
   Since \( X^0 = 1 \), the final answer is:
   \[
   Y^6
   \]

Would you like further clarification or details?

Here are 5 related questions to expand on this topic:
1. How would you simplify \( \frac{X^a Y^b}{X^c Y^d} \times X^e Y^f \)?
2. What happens if the variables \( X \) and \( Y \) have negative exponents?
3. How does the rule of exponents apply to division and multiplication of terms with the same base?
4. How can the product of powers rule be used in solving exponential equations?
5. What is the significance of \( X^0 = 1 \) for any non-zero \( X \)?

**Tip:** When multiplying powers with the same base, always add the exponents.

Learn how to simplify the expression \( \frac{X^2 Y^3}{X^3 Y^{-2}} \times X Y \) step-by-step. This problem covers exponent rules, including product and quotient of powers, and is suitable for students in Grades 8-10. The final solution is \( Y^6 \).

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