Lets simplify each one of the rational expressions using the product rule of exponents: [tex]a^n*a^m=a^{n+m}[/tex] and the quotient rule of exponents: [tex] \frac{a^n}{a^m} =a^{n-m}[/tex].
1. [tex] \frac{-3x^2}{2y^2} * \frac{y^2}{9x} =- \frac{x}{6} = \frac{-1}{6} x[/tex]
2. [tex] \frac{5y^2}{10x^2} * \frac{4x^2}{y} =2y[/tex]
3. [tex] \frac{-9y^2}{5x} * \frac{-10x^2}{3y} =6xy[/tex]
4. [tex] \frac{14x^2}{5y} * \frac{-10y}{7x} =-4x[/tex]
We can conclude that you should pair the rational expressions with their simplest form as follows:
[tex]6xy-----\ \textgreater \ \frac{-9y^2}{5x} * \frac{-10x^2}{3y}[/tex]
[tex]\frac{-1}{6} x-----\ \textgreater \ \frac{-3x^2}{2y^2} * \frac{y^2}{9x} [/tex]
[tex]-4x-----\ \textgreater \ \frac{14x^2}{5y} * \frac{-10y}{7x}[/tex]
[tex]2y-----\ \textgreater \ \frac{5y^2}{10x^2} * \frac{4x^2}{y}[/tex]
