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.