Question

Find each product or quotient.

$\frac{3x-6}{x^2-9}\cdot \frac{x+3}{x^2-2x}$

Short answer

The final product of the expression is $\frac{3}{x\left(x-3\right)}$.

Step-by-step solution

Step 1. Define the concept.

Multiplying rational expression: - Let a, b, c, and d be polynomials with$b\ne 0$ and $d\ne 0$. Then $\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}$.

Dividing rational expression: - Let a, b, c, and d be polynomials with $b\ne 0$,$c\ne 0$ and $d\ne 0$. Then $\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}=\frac{ad}{bc}$.

Step 2. Divide by the common factors before multiplying.

First factor the numerator and denominator of both the rational expressions.

$\frac{3x-6}{x^2-9}\cdot \frac{x+3}{x^2-2x}=\frac{3\left(x-2\right)}{\left(x-3\right)\left(x+3\right)}\cdot \frac{x+3}{x\left(x-2\right)}$

Step 3. Simplify the expression.

Further simplify the expression by dividing by the common factors $x-2$, and $x+3$.

role="math" localid="1648048255063" $\begin{array}{c}\frac{3x-6}{x^2-9}\cdot \frac{x+3}{x^2-2x}=\frac{3\cancel{\left(x-2\right)}}{\left(x-3\right)\cancel{\left(x+3\right)}}\cdot \frac{\cancel{x+3}}{x\cancel{\left(x-2\right)}}\\ =\frac{3}{x-3}\cdot \frac{1}{x}\\ =\frac{3}{x\left(x-3\right)}\end{array}$

Therefore, the final product of the expression is $\frac{3}{x\left(x-3\right)}$.