Let's match each product of rational expressions with their simplest forms.
1. Consider the first expression:
[tex]\[
\frac{-3 x^2}{2 y^2} \times \frac{y^2}{9 x}
\][/tex]
Simplifying step-by-step:
- The [tex]\( y^2 \)[/tex] terms in the numerator and denominator cancel out.
- This leaves:
[tex]\[
\frac{-3 x^2}{2} \times \frac{1}{9 x} = \frac{-3 x^2}{18 x}
\][/tex]
- Simplify by canceling one [tex]\( x \)[/tex]:
[tex]\[
\frac{-3 x}{18} = -\frac{x}{6}
\][/tex]
Therefore, [tex]\(\frac{-3 x^2}{2 y^2} \times \frac{y^2}{9 x} \rightarrow -\frac{x}{6}\)[/tex].
2. Consider the second expression:
[tex]\[
\frac{5 y^2}{10 x^2} \times \frac{4 x^2}{y}
\][/tex]
Simplifying step-by-step:
- The [tex]\( x^2 \)[/tex] terms in the numerator and denominator cancel out.
- This leaves:
[tex]\[
\frac{5 y^2}{10} \times \frac{4}{y} = \frac{5 y^2 \cdot 4}{10 y} = \frac{20 y^2}{10 y}
\][/tex]
- Simplify by canceling one [tex]\( y \)[/tex]:
[tex]\[
\frac{20 y}{10} = 2 y
\][/tex]
Therefore, [tex]\(\frac{5 y^2}{10 x^2} \times \frac{4 x^2}{y} \rightarrow 2 y\)[/tex].
3. Consider the third expression:
[tex]\[
\frac{-9 y^2}{5 x} \times \frac{-10 x^2}{3 y}
\][/tex]
Simplifying step-by-step:
- Combine the expressions:
[tex]\[
\frac{-9 y^2 \times -10 x^2}{5 x \times 3 y} = \frac{90 x^2 y^2}{15 x y}
\][/tex]
- Simplify by canceling one [tex]\( x \)[/tex] and one [tex]\( y \)[/tex]:
[tex]\[
\frac{90 x y}{15} = 6 x y
\][/tex]
Therefore, [tex]\(\frac{-9 y^2}{5 x} \times \frac{-10 x^2}{3 y} \rightarrow 6 x y\)[/tex].
4. Consider the fourth expression:
[tex]\[
\frac{14 x^2}{5 y} \times \frac{-10 y}{7 x}
\][/tex]
Simplifying step-by-step:
- The [tex]\( y \)[/tex] terms in the numerator and denominator cancel out.
- This leaves:
[tex]\[
\frac{14 x^2 \times -10}{5 \times 7 x} = \frac{-140 x^2}{35 x}
\][/tex]
- Simplify by canceling one [tex]\( x \)[/tex]:
[tex]\[
\frac{-140 x}{35} = -4 x
\][/tex]
Therefore, [tex]\(\frac{14 x^2}{5 y} \times \frac{-10 y}{7 x} \rightarrow -4 x\)[/tex].
So we can complete the pairs accordingly:
[tex]\[
\boxed{
\begin{array}{cc}
\frac{-3 x^2}{2 y^2} \times \frac{y^2}{9 x} & \rightarrow -\frac{x}{6} \\
\frac{5 y^2}{10 x^2} \times \frac{4 x^2}{y} & \rightarrow 2 y \\
\frac{-9 y^2}{5 x} \times \frac{-10 x^2}{3 y} & \rightarrow 6 x y \\
\frac{14 x^2}{5 y} \times \frac{-10 y}{7 x} & \rightarrow -4 x \\
\end{array}
}
\][/tex]
