EDGE Math 8

Study how the fractions and rational expressions are multiplied in the examples below.

\frac{2}{3} . \frac{5}{7} = \frac{2 \times 5}{3 \times 7} = \frac{10}{21}

\frac{y}{5 b} . \frac{3}{4 b^{2}} = \frac{y \times 3}{5 b \times 4 b^{2}} = \frac{3 y}{20 b^{3}}

In general, the product of two rational expressions is equal to the product of the numerators over the product of the denominators; that is,

\frac{a}{b} . \frac{c}{d} = \frac{a c}{b d}

Example 1

Multiply and then simplify if possible.

\frac{3 b}{5 a^{3}} . \frac{2}{18 b^{4}}

\frac{x^{2} - 9}{x + 2} . \frac{4 x}{x - 3}

Solution

Use the following rule: $\frac{a}{b} . \frac{c}{d} = \frac{a c}{b d}$

\frac{3 b}{5 a^{3}} . \frac{2}{18 b^{4}} = \frac{3 b (2)}{(5 a^{3}) (18 b^{4})}

= \frac{6 b}{5 a^{3} \cdot 6 b (3 b^{3})}

= \frac{1}{15 a^{3} b^{3}}

b. $\frac{x^{2} - 9}{x + 2} \cdot \frac{4 x}{x - 3} = \frac{(x^{2} - 9) 4 x}{(x + 2) (x - 3)}$

Factor the numerator. Then cancel common factors.

$\frac{(x^{2} - 9) 4 x}{(x + 2) (x - 3)} = \frac{(x + 3) (x - 3) 4 x}{(x + 2) (x - 3)} = \frac{4 x (x + 3)}{x + 2}$