To determine the restricted values for the given expression \(\frac{3xy^2}{4y} \div \frac{2x^2}{3y}\), we need to identify the values of \(x\) and \(y\) that would make any part of the expression undefined.
Let's start by simplifying the expression step-by-step.
1. The given expression is:
[tex]\[
\frac{3xy^2}{4y} \div \frac{2x^2}{3y}
\][/tex]
2. The division of fractions can be transformed into multiplication by the reciprocal of the second fraction:
[tex]\[
\frac{3xy^2}{4y} \times \frac{3y}{2x^2}
\][/tex]
3. Now, simplify each part of the multiplication. Simplify \(\frac{3xy^2}{4y}\):
[tex]\[
\frac{3xy^2}{4y} = \frac{3x \cancel{y^2}}{4 \cancel{y}} = \frac{3xy}{4}
\][/tex]
4. Next, simplify \(\frac{3y}{2x^2}\):
[tex]\[
\frac{3y}{2x^2} \text{ cannot be simplified further.}
\][/tex]
5. Multiply the simplified fractions:
[tex]\[
\frac{3xy}{4} \times \frac{3y}{2x^2} = \frac{(3xy) \cdot (3y)}{4 \cdot 2x^2}
\][/tex]
[tex]\[
= \frac{9xy^2}{8x^2}
\][/tex]
6. Simplify the resulting fraction:
[tex]\[
\frac{9xy^2}{8x^2} = \frac{9 \cancel{xy^2}}{8 \cancel{x^2}} = \frac{9y^2}{8x}
\][/tex]
In the simplified expression \(\frac{9y^2}{8x}\), values of \(x\) and \(y\) that make the denominator zero need to be excluded, as division by zero is undefined.
- For the simplified form \(\frac{9y^2}{8x}\), the denominator is \(8x\). For this to be defined, \(x\) should not be zero.
Additionally, from our initial steps:
- In the original fractions \(\frac{3xy^2}{4y}\) and \(\frac{2x^2}{3y}\), \(y\) should not be zero to avoid division by zero in both fractions.
Therefore, the restricted values are that both \(x\) and \(y\) should not be zero.
The correct answer is:
[tex]\[
\boxed{x \neq 0, y \neq 0}
\][/tex]
