$$ \frac{x^3y^2}{xy^{-3}^{}\cdot2x^0y^3} $$

Let's simplify the given expression step by step:

The expression is:

\[
\frac{x^3y^2}{xy^{-3} \cdot 2x^0y^3}
\]

### Step 1: Simplify the denominator

First, rewrite the denominator $ xy^{-3} \cdot 2x^0y^3 $:
- $ x^0 = 1 $, since anything raised to the power of 0 is 1.
- Now, the denominator becomes $ xy^{-3} \cdot 2y^3 $.

Combine the terms in the denominator:
\[
xy^{-3} \cdot 2y^3 = 2x \cdot y^{-3+3} = 2x.
\]

So, the expression becomes:
\[
\frac{x^3y^2}{2x}.
\]

### Step 2: Simplify the fraction

Now, simplify the fraction:
- For the $ x $-terms: $ \frac{x^3}{x} = x^{3-1} = x^2 $.
- The $ y $-terms remain $ y^2 $, as there are no $ y $-terms in the denominator.

Thus, the expression simplifies to:
\[
\frac{x^2y^2}{2}.
\]

### Final Answer:

\[
\frac{x^2y^2}{2}.
\]

Would you like more details on this solution or have any questions?

Here are five related questions for further exploration:

1. How do you simplify expressions with negative exponents?
2. What happens when an exponent is zero in a term?
3. How can you simplify a fraction where both numerator and denominator have multiple variables?
4. What are the rules for multiplying exponents?
5. How do you simplify fractions that contain both constants and variables?

**Tip:** Always combine like terms and simplify the constants first when dealing with fractions.

$$ \frac{x^3y^2}{xy^{-3} \cdot 2x^0y^3} $$

Learn how to simplify the rational expression $$ \frac{x^3y^2}{xy^{-3} \cdot 2x^0y^3} $$ using exponent rules. This problem covers key algebraic concepts such as simplifying expressions with negative exponents and combining terms. Suitable for students in Grades 8-10.

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