EDGE Math 8

Division of Rational Expressions

Recall that to divide $\frac{10}{7}$ by $\frac{8}{7},$ you can multiply $\frac{10}{7}$ by the reciprocal of $\frac{8}{7},$ which is $\frac{7}{8} .$

\frac{10}{7} \div \frac{8}{7} = \frac{10}{7} \cdot \frac{7}{8} = \frac{10}{8} = \frac{5}{4}

The same procedure is done when dividing rational expressions. Hence, to divide a rational expression, multiply the first rational expression (the dividend) by the reciprocal of the second rational expression (the divisor). In symbols,

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

Example 1

Divide and then simplify if possible.

\frac{3 x}{10} \div \frac{6}{7 y}

Solution

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

\frac{3 x}{10} \div \frac{6}{7 y} = \frac{3 x}{10} \cdot \frac{7 y}{6}

= \frac{3 x \cdot 7 y}{10 \cdot 6}

= \frac{21 x y}{60}

= \frac{7 x y}{20}