# Multiplying Rational Expressions

The method of multiplying rational expressions is same as the method of multiplying fractions . That is, just multiply the numerators to get the numerator of the product and multiply the denominators to get the denominator of the product.

For all rational expressions $\frac{a}{b}$ and $\frac{c}{d}$ with $b\ne 0$ and $d\ne 0$ , $\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}$ .

Example:

Multiply and then simplify.

$\frac{8a}{5b}\cdot \frac{20a{b}^{2}}{16ac}$

Multiply the numerators and multiply the denominators. Arrange the expression so like variables are together.

$\frac{8a}{5b}\cdot \frac{20a{b}^{2}}{16ac}=\frac{\left(8\right)\left(20\right)\left(a\cdot a\right){b}^{2}}{\left(5\right)\left(16\right)abc}$

Divide out the common factors.

$=\frac{\left({}^{2}\overline{)8}\right)\left({\overline{)20}}^{{\overline{)4}}^{1}}\right)\left(\overline{)a}\cdot a\right){\overline{)b}}^{2}}{\left(\overline{)5}\right)\left({\overline{)16}}_{{\overline{)4}}_{1}}\right)\overline{)a}\overline{)b}c}$

Simplify.

$\begin{array}{l}=\frac{2\left(a\right)\left(b\right)}{c}\\ =\frac{2ab}{c}\end{array}$

Note: You can speed up the process by reducing first , before multiplying.