Let's break down the multiplication and simplification of the given expression step by step:
Given expression:
[tex]\[
\frac{4 x^2 y}{x y} \cdot \frac{y^2}{2 x^3}
\][/tex]
### Step 1: Simplify [tex]\(\frac{4 x^2 y}{x y}\)[/tex]
1. In the numerator, [tex]\(4 x^2 y\)[/tex] can be split as:
[tex]\[ 4 \cdot x^2 \cdot y \][/tex]
2. In the denominator, [tex]\(x y\)[/tex] can be split as:
[tex]\[ x \cdot y \][/tex]
3. Cancel the common terms [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in the numerator and denominator:
[tex]\[ \frac{4 x^2 y}{x y} = 4 x \][/tex]
because [tex]\(x^2/x = x\)[/tex] and [tex]\(y/y = 1\)[/tex].
### Step 2: Simplify [tex]\(\frac{y^2}{2 x^3}\)[/tex]
This fraction remains as it is because it is already in its simplest form.
### Step 3: Multiply the simplified fractions
Now, multiply the results from Step 1 and Step 2:
[tex]\[
4 x \cdot \frac{y^2}{2 x^3}
\][/tex]
### Step 4: Perform multiplication
1. Multiply the numerators:
[tex]\[
4 x \cdot y^2 = 4 x y^2
\][/tex]
2. Multiply the denominators:
[tex]\[
1 \cdot 2 x^3 = 2 x^3
\][/tex]
3. Combine the fraction:
[tex]\[
\frac{4 x y^2}{2 x^3}
\][/tex]
### Step 5: Simplify the resulting fraction
1. Divide the numerator and the denominator by the common factor 2:
[tex]\[
\frac{4 x y^2}{2 x^3} = \frac{2 y^2}{x^2}
\][/tex]
### Step 6: Write the final answer
[tex]\[
\boxed{\frac{2 y^2}{x^2}}
\][/tex]
