Contents

Multiply and Divide Rational Expressions

Learning Objectives

By the end of this section, you will be able to:

Multiply rational expressions
Divide rational expressions

Be Prepared 8.2

Before you get started, take this readiness quiz.

If you miss a problem, go back to the section listed and review the material.

Multiply: $\frac{14}{15} \cdot \frac{6}{35} . \frac{14}{15} \cdot \frac{6}{35} .$
If you missed this problem, review Example 1.68.
Divide: $\frac{14}{15} \div \frac{6}{35} . \frac{14}{15} \div \frac{6}{35} .$
If you missed this problem, review Example 1.71.
Factor completely: $2 x^{2} - 98 . 2 x^{2} - 98 .$
If you missed this problem, review Example 7.62.
Factor completely: $10 n^{3} + 10 . 10 n^{3} + 10 .$
If you missed this problem, review Example 7.65.
Factor completely: $10 p^{2} - 25 p q - 15 q^{2} . 10 p^{2} - 25 p q - 15 q^{2} .$
If you missed this problem, review Example 7.68.

Multiply Rational Expressions

To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.

Multiplication of Rational Expressions

If $p, q, r, s p, q, r, s$ are polynomials where $q \neq 0 and s \neq 0 q \neq 0$ , then

\frac{p}{q} \cdot \frac{r}{s} = \frac{p r}{q s} \frac{p}{q} \cdot \frac{r}{s} = \frac{p r}{q s}

To multiply rational expressions, multiply the numerators and multiply the denominators.

We’ll do the first example with numerical fractions to remind us of how we multiplied fractions without variables.

Example 8.17

Multiply: $\frac{10}{28} \cdot \frac{8}{15} . \frac{10}{28} \cdot \frac{8}{15} .$

Show solution

Solution


Multiply the numerators and denominators.
Look for common factors, and then remove them.
Simplify.

Try It 8.33

Mulitply: $\frac{6}{10} \cdot \frac{15}{12} . \frac{6}{10} \cdot \frac{15}{12} .$

Try It 8.34

Mulitply: $\frac{20}{15} \cdot \frac{6}{8} . \frac{20}{15} \cdot \frac{6}{8} .$

Remember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, $x \neq 0 x \neq 0$ and $y \neq 0 y \neq 0$ .

Example 8.18

Mulitply: $\frac{2 x}{3 y^{2}} \cdot \frac{6 x y^{3}}{x^{2} y} . \frac{2 x}{3 y^{2}} \cdot \frac{6 x y^{3}}{x^{2} y} .$

Show solution

Solution


Multiply.
Factor the numerator and denominator completely, and then remove common factors.
Simplify.

Try It 8.35

Mulitply: $\frac{3 p q}{q^{2}} \cdot \frac{5 p^{2} q}{6 p q} . \frac{3 p q}{q^{2}} \cdot \frac{5 p^{2} q}{6 p q} .$

Try It 8.36

Mulitply: $\frac{6 x^{3} y}{7 x^{2}} \cdot \frac{2 x y^{3}}{x^{2} y} . \frac{6 x^{3} y}{7 x^{2}} \cdot \frac{2 x y^{3}}{x^{2} y} .$

Example 8.19

How to Multiply Rational Expressions

Mulitply: $\frac{2 x}{x^{2} - 7 x + 12} \cdot \frac{x^{2} - 9}{6 x^{2}} . \frac{2 x}{x^{2} - 7 x + 12} \cdot \frac{x^{2} - 9}{6 x^{2}} .$

Show solution

Solution

The above image has three columns and three rows to show how to multiply rational expressions. Step one is to factor each numerator and denominator completely. Factor x squared minus 9 and x squared plus x plus 12. The rational equation is 2x divided by x squared plus x plus 12 times x squared minus 9 divided by 6x squared, then to 2x divided by x minus 3 times x minus 4 times x minus 3 times x plus 3 divided by 6x squared.

Step 2 is to multiply the numerators and denominators. It is helpful to multiply the monomials first. Multiply 2x times x minus 3 times x plus 3 divided by 6x squared times x minus 3 times x minus 4.

Step 3 is to divide out the common factors, canceling out 2, x, and x minus 3 in the numerator and 2, x and x minus 3 in the denominator. Leave the denominator in factored form to get x plus 3 divided by 3x times x minus 4.

Try It 8.37

Mulitply: $\frac{5 x}{x^{2} + 5 x + 6} \cdot \frac{x^{2} - 4}{10 x} . \frac{5 x}{x^{2} + 5 x + 6} \cdot \frac{x^{2} - 4}{10 x} .$

Try It 8.38

Mulitply: $\frac{9 x^{2}}{x^{2} + 11 x + 30} \cdot \frac{x^{2} - 36}{3 x^{2}} . \frac{9 x^{2}}{x^{2} + 11 x + 30} \cdot \frac{x^{2} - 36}{3 x^{2}} .$

How To

Multiply a rational expression.

Step 1. Factor each numerator and denominator completely.
Step 2. Multiply the numerators and denominators.
Step 3. Simplify by dividing out common factors.

Example 8.20

Multiply: $\frac{n^{2} - 7 n}{n^{2} + 2 n + 1} \cdot \frac{n + 1}{2 n} . \frac{n^{2} - 7 n}{n^{2} + 2 n + 1} \cdot \frac{n + 1}{2 n} .$

Show solution

Solution

$\begin{matrix} \frac{n^{2} - 7 n}{n^{2} + 2 n + 1} \cdot \frac{n + 1}{2 n} \\ Factor each numerator and denominator. & \frac{n (n - 7)}{(n + 1) (n + 1)} \cdot \frac{n + 1}{2 n} \\ \begin{matrix} Multiply the numerators and the \\ denominators. \end{matrix} & \frac{n (n - 7) (n + 1)}{(n + 1) (n + 1) 2 n} \\ Remove common factors. & \frac{n (n - 7) (n + 1)}{(n + 1) (n + 1) 2 n} \\ Simplify. & \frac{n - 7}{2 (n + 1)} \end{matrix} \begin{matrix} Factor each numerator and denominator. & \begin{matrix} Multiply the numerators and the \\ denominators. \end{matrix} & Remove common factors. & Simplify. \end{matrix}$

Try It 8.39

Multiply: $\frac{x^{2} - 25}{x^{2} - 3 x - 10} \cdot \frac{x + 2}{x} . \frac{x^{2} - 25}{x^{2} - 3 x - 10} \cdot \frac{x + 2}{x} .$

Try It 8.40

Multiply: $\frac{x^{2} - 4 x}{x^{2} + 5 x + 6} \cdot \frac{x + 2}{x} . \frac{x^{2} - 4 x}{x^{2} + 5 x + 6} \cdot \frac{x + 2}{x} .$

Example 8.21

Multiply: $\frac{16 - 4 x}{2 x - 12} \cdot \frac{x^{2} - 5 x - 6}{x^{2} - 16} . \frac{16 - 4 x}{2 x - 12} \cdot \frac{x^{2} - 5 x - 6}{x^{2} - 16} .$

Show solution

Solution

$\begin{matrix} \frac{16 - 4 x}{2 x - 12} \cdot \frac{x^{2} - 5 x - 6}{x^{2} - 16} \\ Factor each numerator and denominator. & \frac{4 (4 - x)}{2 (x - 6)} \cdot \frac{(x - 6) (x + 1)}{(x - 4) (x + 4)} \\ \begin{matrix} Multiply the numerators and the \\ denominators. \end{matrix} & \frac{4 (4 - x) (x - 6) (x + 1)}{2 (x - 6) (x - 4) (x + 4)} \\ Remove common factors. & (−1) \frac{2 \cdot 2 (4 - x) (x - 6) (x + 1)}{2 (x - 6) (x - 4) (x + 4)} \\ Simplify. & - \frac{2 (x + 1)}{(x + 4)} \end{matrix} \begin{matrix} Factor each numerator and denominator. & \begin{matrix} Multiply the numerators and the \\ denominators. \end{matrix} & Remove common factors. & Simplify. \end{matrix}$

Try It 8.41

Multiply: $\frac{12 x - 6 x^{2}}{x^{2} + 8 x} \cdot \frac{x^{2} + 11 x + 24}{x^{2} - 4} . \frac{12 x - 6 x^{2}}{x^{2} + 8 x} \cdot \frac{x^{2} + 11 x + 24}{x^{2} - 4} .$

Try It 8.42

Multiply: $\frac{9 v - 3 v^{2}}{9 v + 36} \cdot \frac{v^{2} + 7 v + 12}{v^{2} - 9} . \frac{9 v - 3 v^{2}}{9 v + 36} \cdot \frac{v^{2} + 7 v + 12}{v^{2} - 9} .$

Example 8.22

Multiply: $\frac{2 x - 6}{x^{2} - 8 x + 15} \cdot \frac{x^{2} - 25}{2 x + 10} . \frac{2 x - 6}{x^{2} - 8 x + 15} \cdot \frac{x^{2} - 25}{2 x + 10} .$

Show solution

Solution


Factor each numerator and denominator.
Multiply the numerators and denominators.
Remove common factors.
Simplify.

Try It 8.43

Multiply: $\frac{3 a - 21}{a^{2} - 9 a + 14} \cdot \frac{a^{2} - 4}{3 a + 6} . \frac{3 a - 21}{a^{2} - 9 a + 14} \cdot \frac{a^{2} - 4}{3 a + 6} .$

Try It 8.44

Multiply: $\frac{b^{2} - b}{b^{2} + 9 b - 10} \cdot \frac{b^{2} - 100}{b^{2} - 10 b} . \frac{b^{2} - b}{b^{2} + 9 b - 10} \cdot \frac{b^{2} - 100}{b^{2} - 10 b} .$

Divide Rational Expressions

To divide rational expressions we multiply the first fraction by the reciprocal of the second, just like we did for numerical fractions.

Remember, the reciprocal of $\frac{a}{b} \frac{a}{b}$ is $\frac{b}{a} \frac{b}{a}$ . To find the reciprocal we simply put the numerator in the denominator and the denominator in the numerator. We “flip” the fraction.

Division of Rational Expressions

If $p, q, r, s p, q, r, s$ are polynomials where $q \neq 0, r \neq 0, s \neq 0 q \neq 0, r \neq 0, s \neq 0$ , then

\frac{p}{q} \div \frac{r}{s} = \frac{p}{q} \cdot \frac{s}{r} \frac{p}{q} \div \frac{r}{s} = \frac{p}{q} \cdot \frac{s}{r}

To divide rational expressions multiply the first fraction by the reciprocal of the second.

Example 8.23

How to Divide Rational Expressions

Divide: $\frac{x + 9}{6 - x} \div \frac{x^{2} - 81}{x - 6} . \frac{x + 9}{6 - x} \div \frac{x^{2} - 81}{x - 6} .$

Show solution

Solution

$The above image has three columns. It shows the steps to divide rational expressions. Step one is to rewrite the division as the product of the first rational expression and the reciprocal of the second for x plus 9 divided by 6 minus x divided by x squared minus 81 divided by x minus 6. “Flip” the second fraction and change the division sign to multiplication to get x plus 9 divided by 6 minus x times x minus 6 divided by x squared minus 81.$ Step two is to factor the numerators and denominators completely. Factor x squared minus 81 to get x plus 9 divided by 6 minus x times x minus 6 divided by x minus 9 times x plus 9.

Step two is to factor the numerators and denominators completely. Factor x squared minus 81 to get x plus 9 divided by 6 minus x times x minus 6 divided by x minus 9 times x plus 9.

Step three is to multiply the numerators and denominators to get x plus 9 times x minus 6 divided by 6 minus x times x minus 9 times x plus 9.

Step four is to simplify by dividing out common factors. Divide out the common factors x plus 9, x minus 6 from the numerator and 6 minus x and x plus 9 from the denominator. Remember opposites divide to negative 1. This simplifies to negative 1 divided by x minus 9.

Try It 8.45

Divide: $\frac{c + 3}{5 - c} \div \frac{c^{2} - 9}{c - 5} . \frac{c + 3}{5 - c} \div \frac{c^{2} - 9}{c - 5} .$

Try It 8.46

Divide: $\frac{2 - d}{d - 4} \div \frac{4 - d^{2}}{4 - d} . \frac{2 - d}{d - 4} \div \frac{4 - d^{2}}{4 - d} .$

How To

Divide rational expressions.

Step 1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
Step 2. Factor the numerators and denominators completely.
Step 3. Multiply the numerators and denominators together.
Step 4. Simplify by dividing out common factors.

Example 8.24

Divide: $\frac{3 n^{2}}{n^{2} - 4 n} \div \frac{9 n^{2} - 45 n}{n^{2} - 7 n + 10} . \frac{3 n^{2}}{n^{2} - 4 n} \div \frac{9 n^{2} - 45 n}{n^{2} - 7 n + 10} .$

Show solution

Solution


Rewrite the division as the product of the first rational expression and the reciprocal of the second.
Factor the numerators and denominators and then multiply.
Simplify by dividing out common factors.

Try It 8.47

Divide: $\frac{2 m^{2}}{m^{2} - 8 m} \div \frac{8 m^{2} + 24 m}{m^{2} + m - 6} . \frac{2 m^{2}}{m^{2} - 8 m} \div \frac{8 m^{2} + 24 m}{m^{2} + m - 6} .$

Try It 8.48

Divide: $\frac{15 n^{2}}{3 n^{2} + 33 n} \div \frac{5 n - 5}{n^{2} + 9 n - 22} . \frac{15 n^{2}}{3 n^{2} + 33 n} \div \frac{5 n - 5}{n^{2} + 9 n - 22} .$

Remember, first rewrite the division as multiplication of the first expression by the reciprocal of the second. Then factor everything and look for common factors.

Example 8.25

Divide: $\frac{2 x^{2} + 5 x - 12}{x^{2} - 16} \div \frac{2 x^{2} - 13 x + 15}{x^{2} - 8 x + 16} . \frac{2 x^{2} + 5 x - 12}{x^{2} - 16} \div \frac{2 x^{2} - 13 x + 15}{x^{2} - 8 x + 16} .$

Show solution

Solution

$\begin{matrix} \frac{2 x^{2} + 5 x - 12}{x^{2} - 16} \div \frac{2 x^{2} - 13 x + 15}{x^{2} - 8 x + 16} \\ \begin{matrix} Rewrite the division as multiplication of \\ the first expression by the reciprocal of \\ the second. \end{matrix} & \frac{2 x^{2} + 5 x - 12}{x^{2} - 16} \cdot \frac{x^{2} - 8 x + 16}{2 x^{2} - 13 x + 15} \\ \begin{matrix} Factor the numerators and denominators \\ and then multiply. \end{matrix} & \frac{(2 x - 3) (x + 4) (x - 4) (x - 4)}{(x - 4) (x + 4) (2 x - 3) (x - 5)} \\ Simplify by dividing out common factors. & \frac{(2 x - 3) (x + 4) (x - 4) (x - 4)}{(x - 4) (x + 4) (2 x - 3) (x - 5)} \\ Simplify. & \frac{x - 4}{x - 5} \end{matrix} \begin{matrix} \begin{matrix} Rewrite the division as multiplication of \\ the first expression by the reciprocal of \\ the second. \end{matrix} & \begin{matrix} Factor the numerators and denominators \\ and then multiply. \end{matrix} & Simplify by dividing out common factors. & Simplify. \end{matrix}$

Try It 8.49

Divide: $\frac{3 a^{2} - 8 a - 3}{a^{2} - 25} \div \frac{3 a^{2} - 14 a - 5}{a^{2} + 10 a + 25} . \frac{3 a^{2} - 8 a - 3}{a^{2} - 25} \div \frac{3 a^{2} - 14 a - 5}{a^{2} + 10 a + 25} .$

Try It 8.50

Divide: $\frac{4 b^{2} + 7 b - 2}{1 - b^{2}} \div \frac{4 b^{2} + 15 b - 4}{b^{2} - 2 b + 1} . \frac{4 b^{2} + 7 b - 2}{1 - b^{2}} \div \frac{4 b^{2} + 15 b - 4}{b^{2} - 2 b + 1} .$

Example 8.26

Divide: $\frac{p^{3} + q^{3}}{2 p^{2} + 2 p q + 2 q^{2}} \div \frac{p^{2} - q^{2}}{6} . \frac{p^{3} + q^{3}}{2 p^{2} + 2 p q + 2 q^{2}} \div \frac{p^{2} - q^{2}}{6} .$

Show solution

Solution

$\begin{matrix} \frac{p^{3} + q^{3}}{2 p^{2} + 2 p q + 2 q^{2}} \div \frac{p^{2} - q^{2}}{6} \\ \begin{matrix} Rewrite the division as a multiplication \\ of the first expression times the \\ reciprocal of the second. \end{matrix} & \frac{p^{3} + q^{3}}{2 p^{2} + 2 p q + 2 q^{2}} \cdot \frac{6}{p^{2} - q^{2}} \\ \begin{matrix} Factor the numerators and denominators \\ and then multiply. \end{matrix} & \frac{(p + q) (p^{2} - p q + q^{2}) 6}{2 (p^{2} + p q + q^{2}) (p - q) (p + q)} \\ Simplify by dividing out common factors. & \frac{(p + q) (p^{2} - p q + q^{2}) 6^{3}}{2 (p^{2} + p q + q^{2}) (p - q) (p + q)} \\ Simplify. & \frac{3 (p^{2} - p q + q^{2})}{(p - q) (p^{2} + p q + q^{2})} \end{matrix} \begin{matrix} \begin{matrix} Rewrite the division as a multiplication \\ of the first expression times the \\ reciprocal of the second. \end{matrix} & \begin{matrix} Factor the numerators and denominators \\ and then multiply. \end{matrix} & Simplify by dividing out common factors. & Simplify. \end{matrix}$

Try It 8.51

Divide: $\frac{x^{3} - 8}{3 x^{2} - 6 x + 12} \div \frac{x^{2} - 4}{6} . \frac{x^{3} - 8}{3 x^{2} - 6 x + 12} \div \frac{x^{2} - 4}{6} .$

Try It 8.52

Divide: $\frac{2 z^{2}}{z^{2} - 1} \div \frac{z^{3} - z^{2} + z}{z^{3} - 1} . \frac{2 z^{2}}{z^{2} - 1} \div \frac{z^{3} - z^{2} + z}{z^{3} - 1} .$

Before doing the next example, let’s look at how we divide a fraction by a whole number. When we divide $\frac{3}{5} \div 4 \frac{3}{5} \div 4$ , we first write 4 as a fraction so that we can find its reciprocal.

\begin{matrix} \frac{3}{5} \div 4 \\ \frac{3}{5} \div \frac{4}{1} \\ \frac{3}{5} \cdot \frac{1}{4} \end{matrix} \begin{matrix} \frac{3}{5} \div 4 \\ \frac{3}{5} \div \frac{4}{1} \\ \frac{3}{5} \cdot \frac{1}{4} \end{matrix}

We do the same thing when we divide rational expressions.

Example 8.27

Divide: $\frac{a^{2} - b^{2}}{3 a b} \div (a^{2} + 2 a b + b^{2}) . \frac{a^{2} - b^{2}}{3 a b} \div (a^{2} + 2 a b + b^{2}) .$

Show solution

Solution

$\begin{matrix} \frac{a^{2} - b^{2}}{3 a b} \div (a^{2} + 2 a b + b^{2}) \\ Write the second expression as a fraction. & \frac{a^{2} - b^{2}}{3 a b} \div \frac{a^{2} + 2 a b + b^{2}}{1} \\ \begin{matrix} Rewrite the division as the first \\ expression times the reciprocal of the \\ second expression. \end{matrix} & \frac{a^{2} - b^{2}}{3 a b} \cdot \frac{1}{a^{2} + 2 a b + b^{2}} \\ \begin{matrix} Factor the numerators and the \\ denominators, and then multiply. \end{matrix} & \frac{(a - b) (a + b) \cdot 1}{3 a b \cdot (a + b) (a + b)} \\ Simplify by dividing out common factors. & \frac{(a - b) (a + b)}{3 a b \cdot (a + b) (a + b)} \\ Simplify. & \frac{(a - b)}{3 a b (a + b)} \end{matrix} \begin{matrix} Write the second expression as a fraction. & \begin{matrix} Rewrite the division as the first \\ expression times the reciprocal of the \\ second expression. \end{matrix} & \begin{matrix} Factor the numerators and the \\ denominators, and then multiply. \end{matrix} & Simplify by dividing out common factors. & Simplify. \end{matrix}$

Try It 8.53

Divide: $\frac{2 x^{2} - 14 x - 16}{4} \div (x^{2} + 2 x + 1) . \frac{2 x^{2} - 14 x - 16}{4} \div (x^{2} + 2 x + 1) .$

Try It 8.54

Divide: $\frac{y^{2} - 6 y + 8}{y^{2} - 4 y} \div (3 y^{2} - 12 y) . \frac{y^{2} - 6 y + 8}{y^{2} - 4 y} \div (3 y^{2} - 12 y) .$

Remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.

Example 8.28

Divide: $\frac{\frac{6 x^{2} - 7 x + 2}{4 x - 8}}{\frac{2 x^{2} - 7 x + 3}{x^{2} - 5 x + 6}} . \frac{\frac{6 x^{2} - 7 x + 2}{4 x - 8}}{\frac{2 x^{2} - 7 x + 3}{x^{2} - 5 x + 6}} .$

Show solution

Solution

$\begin{matrix} \frac{\frac{6 x^{2} - 7 x + 2}{4 x - 8}}{\frac{2 x^{2} - 7 x + 3}{x^{2} - 5 x + 6}} \\ Rewrite with a division sign. & \frac{6 x^{2} - 7 x + 2}{4 x - 8} \div \frac{2 x^{2} - 7 x + 3}{x^{2} - 5 x + 6} \\ \begin{matrix} Rewrite as product of first times \\ reciprocal of second. \end{matrix} & \frac{6 x^{2} - 7 x + 2}{4 x - 8} \cdot \frac{x^{2} - 5 x + 6}{2 x^{2} - 7 x + 3} \\ \begin{matrix} Factor the numerators and the \\ denominators, and then multiply. \end{matrix} & \frac{(2 x - 1) (3 x - 2) (x - 2) (x - 3)}{4 (x - 2) (2 x - 1) (x - 3)} \\ Simplify by dividing out common factors. & \frac{(2 x - 1) (3 x - 2) (x - 2) (x - 3)}{4 (x - 2) (2 x - 1) (x - 3)} \\ Simplify. & \frac{3 x - 2}{4} \end{matrix} \begin{matrix} Rewrite with a division sign. & \begin{matrix} Rewrite as product of first times \\ reciprocal of second. \end{matrix} & \begin{matrix} Factor the numerators and the \\ denominators, and then multiply. \end{matrix} & Simplify by dividing out common factors. & Simplify. \end{matrix}$

Try It 8.55

Divide: $\frac{\frac{3 x^{2} + 7 x + 2}{4 x + 24}}{\frac{3 x^{2} - 14 x - 5}{x^{2} + x - 30}} . \frac{\frac{3 x^{2} + 7 x + 2}{4 x + 24}}{\frac{3 x^{2} - 14 x - 5}{x^{2} + x - 30}} .$

Try It 8.56

Divide: $\frac{\frac{y^{2} - 36}{2 y^{2} + 11 y - 6}}{\frac{2 y^{2} - 2 y - 60}{8 y - 4}} . \frac{\frac{y^{2} - 36}{2 y^{2} + 11 y - 6}}{\frac{2 y^{2} - 2 y - 60}{8 y - 4}} .$

If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then we factor and multiply.

Example 8.29

Divide: $\frac{3 x - 6}{4 x - 4} \cdot \frac{x^{2} + 2 x - 3}{x^{2} - 3 x - 10} \div \frac{2 x + 12}{8 x + 16} . \frac{3 x - 6}{4 x - 4} \cdot \frac{x^{2} + 2 x - 3}{x^{2} - 3 x - 10} \div \frac{2 x + 12}{8 x + 16} .$

Show solution

Solution


Rewrite the division as multiplication by the reciprocal.
Factor the numerators and the denominators, and then multiply.
Simplify by dividing out common factors.
Simplify.

Try It 8.57

Divide: $\frac{4 m + 4}{3 m - 15} \cdot \frac{m^{2} - 3 m - 10}{m^{2} - 4 m - 32} \div \frac{12 m - 36}{6 m - 48} . \frac{4 m + 4}{3 m - 15} \cdot \frac{m^{2} - 3 m - 10}{m^{2} - 4 m - 32} \div \frac{12 m - 36}{6 m - 48} .$

Try It 8.58

Divide: $\frac{2 n^{2} + 10 n}{n - 1} \div \frac{n^{2} + 10 n + 24}{n^{2} + 8 n - 9} \cdot \frac{n + 4}{8 n^{2} + 12 n} . \frac{2 n^{2} + 10 n}{n - 1} \div \frac{n^{2} + 10 n + 24}{n^{2} + 8 n - 9} \cdot \frac{n + 4}{8 n^{2} + 12 n} .$

Practice Makes Perfect

Multiply Rational Expressions

In the following exercises, multiply.

73.

$\frac{12}{16} \cdot \frac{4}{10} \frac{12}{16} \cdot \frac{4}{10}$

74.

$\frac{32}{5} \cdot \frac{16}{24} \frac{32}{5} \cdot \frac{16}{24}$

75.

$\frac{18}{10} \cdot \frac{4}{30} \frac{18}{10} \cdot \frac{4}{30}$

76.

$\frac{21}{36} \cdot \frac{45}{24} \frac{21}{36} \cdot \frac{45}{24}$

77.

$\frac{5 x^{2} y^{4}}{12 x y^{3}} \cdot \frac{6 x^{2}}{20 y^{2}} \frac{5 x^{2} y^{4}}{12 x y^{3}} \cdot \frac{6 x^{2}}{20 y^{2}}$

78.

$\frac{8 w^{3} y}{9 y^{2}} \cdot \frac{3 y}{4 w^{4}} \frac{8 w^{3} y}{9 y^{2}} \cdot \frac{3 y}{4 w^{4}}$

79.

$\frac{12 a^{3} b}{b^{2}} \cdot \frac{2 a b^{2}}{9 b^{3}} \frac{12 a^{3} b}{b^{2}} \cdot \frac{2 a b^{2}}{9 b^{3}}$

80.

$\frac{4 m n^{2}}{5 n^{3}} \cdot \frac{m n^{3}}{8 m^{2} n^{2}} \frac{4 m n^{2}}{5 n^{3}} \cdot \frac{m n^{3}}{8 m^{2} n^{2}}$

81.

$\frac{5 p^{2}}{p^{2} - 5 p - 36} \cdot \frac{p^{2} - 16}{10 p} \frac{5 p^{2}}{p^{2} - 5 p - 36} \cdot \frac{p^{2} - 16}{10 p}$

82.

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

83.

$\frac{4 r}{r^{2} - 3 r - 10} \cdot \frac{r^{2} - 25}{8 r^{2}} \frac{4 r}{r^{2} - 3 r - 10} \cdot \frac{r^{2} - 25}{8 r^{2}}$

84.

$\frac{s}{s^{2} - 9 s + 14} \cdot \frac{s^{2} - 49}{7 s^{2}} \frac{s}{s^{2} - 9 s + 14} \cdot \frac{s^{2} - 49}{7 s^{2}}$

85.

$\frac{x^{2} - 7 x}{x^{2} + 6 x + 9} \cdot \frac{x + 3}{4 x} \frac{x^{2} - 7 x}{x^{2} + 6 x + 9} \cdot \frac{x + 3}{4 x}$

86.

$\frac{2 y^{2} - 10 y}{y^{2} + 10 y + 25} \cdot \frac{y + 5}{6 y} \frac{2 y^{2} - 10 y}{y^{2} + 10 y + 25} \cdot \frac{y + 5}{6 y}$

87.

$\frac{z^{2} + 3 z}{z^{2} - 3 z - 4} \cdot \frac{z - 4}{z^{2}} \frac{z^{2} + 3 z}{z^{2} - 3 z - 4} \cdot \frac{z - 4}{z^{2}}$

88.

$\frac{2 a^{2} + 8 a}{a^{2} - 9 a + 20} \cdot \frac{a - 5}{a^{2}} \frac{2 a^{2} + 8 a}{a^{2} - 9 a + 20} \cdot \frac{a - 5}{a^{2}}$

89.

$\frac{28 - 4 b}{3 b - 3} \cdot \frac{b^{2} + 8 b - 9}{b^{2} - 49} \frac{28 - 4 b}{3 b - 3} \cdot \frac{b^{2} + 8 b - 9}{b^{2} - 49}$

90.

$\frac{18 c - 2 c^{2}}{6 c + 30} \cdot \frac{c^{2} + 7 c + 10}{c^{2} - 81} \frac{18 c - 2 c^{2}}{6 c + 30} \cdot \frac{c^{2} + 7 c + 10}{c^{2} - 81}$

91.

$\frac{35 d - 7 d^{2}}{d^{2} + 7 d} \cdot \frac{d^{2} + 12 d + 35}{d^{2} - 25} \frac{35 d - 7 d^{2}}{d^{2} + 7 d} \cdot \frac{d^{2} + 12 d + 35}{d^{2} - 25}$

92.

$\frac{72 m - 12 m^{2}}{8 m + 32} \cdot \frac{m^{2} + 10 m + 24}{m^{2} - 36} \frac{72 m - 12 m^{2}}{8 m + 32} \cdot \frac{m^{2} + 10 m + 24}{m^{2} - 36}$

93.

$\frac{4 n + 20}{n^{2} + n - 20} \cdot \frac{n^{2} - 16}{4 n + 16} \frac{4 n + 20}{n^{2} + n - 20} \cdot \frac{n^{2} - 16}{4 n + 16}$

94.

$\frac{6 p^{2} - 6 p}{p^{2} + 7 p - 18} \cdot \frac{p^{2} - 81}{3 p^{2} - 27 p} \frac{6 p^{2} - 6 p}{p^{2} + 7 p - 18} \cdot \frac{p^{2} - 81}{3 p^{2} - 27 p}$

95.

$\frac{q^{2} - 2 q}{q^{2} + 6 q - 16} \cdot \frac{q^{2} - 64}{q^{2} - 8 q} \frac{q^{2} - 2 q}{q^{2} + 6 q - 16} \cdot \frac{q^{2} - 64}{q^{2} - 8 q}$

96.

$\frac{2 r^{2} - 2 r}{r^{2} + 4 r - 5} \cdot \frac{r^{2} - 25}{2 r^{2} - 10 r} \frac{2 r^{2} - 2 r}{r^{2} + 4 r - 5} \cdot \frac{r^{2} - 25}{2 r^{2} - 10 r}$

Divide Rational Expressions

In the following exercises, divide.

97.

$\frac{t - 6}{3 - t} \div \frac{t^{2} - 9}{t - 5} \frac{t - 6}{3 - t} \div \frac{t^{2} - 9}{t - 5}$

98.

$\frac{v - 5}{11 - v} \div \frac{v^{2} - 25}{v - 11} \frac{v - 5}{11 - v} \div \frac{v^{2} - 25}{v - 11}$

99.

$\frac{10 + w}{w - 8} \div \frac{100 - w^{2}}{8 - w} \frac{10 + w}{w - 8} \div \frac{100 - w^{2}}{8 - w}$

100.

$\frac{7 + x}{x - 6} \div {\frac{49 - x}{x + 6}}^{2} \frac{7 + x}{x - 6} \div {\frac{49 - x}{x + 6}}^{2}$

101.

$\frac{27 y^{2}}{3 y - 21} \div \frac{3 y^{2} + 18}{y^{2} + 13 y + 42} \frac{27 y^{2}}{3 y - 21} \div \frac{3 y^{2} + 18}{y^{2} + 13 y + 42}$

102.

$\frac{24 z^{2}}{2 z - 8} \div \frac{4 z - 28}{z^{2} - 11 z + 28} \frac{24 z^{2}}{2 z - 8} \div \frac{4 z - 28}{z^{2} - 11 z + 28}$

103.

$\frac{16 a^{2}}{4 a + 36} \div \frac{4 a^{2} - 24 a}{a^{2} + 4 a - 45} \frac{16 a^{2}}{4 a + 36} \div \frac{4 a^{2} - 24 a}{a^{2} + 4 a - 45}$

104.

$\frac{24 b^{2}}{2 b - 4} \div \frac{12 b^{2} + 36 b}{b^{2} - 11 b + 18} \frac{24 b^{2}}{2 b - 4} \div \frac{12 b^{2} + 36 b}{b^{2} - 11 b + 18}$

105.

$\frac{5 c^{2} + 9 c + 2}{c^{2} - 25} \div \frac{3 c^{2} - 14 c - 5}{c^{2} + 10 c + 25} \frac{5 c^{2} + 9 c + 2}{c^{2} - 25} \div \frac{3 c^{2} - 14 c - 5}{c^{2} + 10 c + 25}$

106.

$\frac{2 d^{2} + d - 3}{d^{2} - 16} \div \frac{2 d^{2} - 9 d - 18}{d^{2} - 8 d + 16} \frac{2 d^{2} + d - 3}{d^{2} - 16} \div \frac{2 d^{2} - 9 d - 18}{d^{2} - 8 d + 16}$

107.

$\frac{6 m^{2} - 2 m - 10}{9 - m^{2}} \div \frac{6 m^{2} + 29 m - 20}{m^{2} - 6 m + 9} \frac{6 m^{2} - 2 m - 10}{9 - m^{2}} \div \frac{6 m^{2} + 29 m - 20}{m^{2} - 6 m + 9}$

108.

$\frac{2 n^{2} - 3 n - 14}{25 - n^{2}} \div \frac{2 n^{2} - 13 n + 21}{n^{2} - 10 n + 25} \frac{2 n^{2} - 3 n - 14}{25 - n^{2}} \div \frac{2 n^{2} - 13 n + 21}{n^{2} - 10 n + 25}$

109.

$\frac{3 s^{2}}{s^{2} - 16} \div \frac{s^{3} + 4 s^{2} + 16 s}{s^{3} - 64} \frac{3 s^{2}}{s^{2} - 16} \div \frac{s^{3} + 4 s^{2} + 16 s}{s^{3} - 64}$

110.

$\frac{r^{2} - 9}{15} \div \frac{r^{3} - 27}{5 r^{2} + 15 r + 45} \frac{r^{2} - 9}{15} \div \frac{r^{3} - 27}{5 r^{2} + 15 r + 45}$

111.

$\frac{p^{3} + q^{3}}{3 p^{2} + 3 p q + 3 q^{2}} \div \frac{p^{2} - q^{2}}{12} \frac{p^{3} + q^{3}}{3 p^{2} + 3 p q + 3 q^{2}} \div \frac{p^{2} - q^{2}}{12}$

112.

$\frac{v^{3} - 8 w^{3}}{2 v^{2} + 4 v w + 8 w^{2}} \div \frac{v^{2} - 4 w^{2}}{4} \frac{v^{3} - 8 w^{3}}{2 v^{2} + 4 v w + 8 w^{2}} \div \frac{v^{2} - 4 w^{2}}{4}$

113.

$\frac{t^{2} - 9}{2 t} \div (t^{2} - 6 t + 9) \frac{t^{2} - 9}{2 t} \div (t^{2} - 6 t + 9)$

114.

$\frac{x^{2} + 3 x - 10}{4 x} \div (2 x^{2} + 20 x + 50) \frac{x^{2} + 3 x - 10}{4 x} \div (2 x^{2} + 20 x + 50)$

115.

$\frac{2 y^{2} - 10 y z - 48 z^{2}}{2 y - 1} \div (4 y^{2} - 32 y z) \frac{2 y^{2} - 10 y z - 48 z^{2}}{2 y - 1} \div (4 y^{2} - 32 y z)$

116.

$\frac{2 m^{2} - 98 n^{2}}{2 m + 6} \div (m^{2} - 7 m n) \frac{2 m^{2} - 98 n^{2}}{2 m + 6} \div (m^{2} - 7 m n)$

117.

$\frac{\frac{2 a^{2} - a - 21}{5 a + 20}}{\frac{a^{2} + 7 a + 12}{a^{2} + 8 a + 16}} \frac{\frac{2 a^{2} - a - 21}{5 a + 20}}{\frac{a^{2} + 7 a + 12}{a^{2} + 8 a + 16}}$

118.

$\frac{\frac{3 b^{2} + 2 b - 8}{12 b + 18}}{\frac{3 b^{2} + 2 b - 8}{2 b^{2} - 7 b - 15}} \frac{\frac{3 b^{2} + 2 b - 8}{12 b + 18}}{\frac{3 b^{2} + 2 b - 8}{2 b^{2} - 7 b - 15}}$

119.

$\frac{\frac{12 c^{2} - 12}{2 c^{2} - 3 c + 1}}{\frac{4 c + 4}{6 c^{2} - 13 c + 5}} \frac{\frac{12 c^{2} - 12}{2 c^{2} - 3 c + 1}}{\frac{4 c + 4}{6 c^{2} - 13 c + 5}}$

120.

$\frac{\frac{4 d^{2} + 7 d - 2}{35 d + 10}}{\frac{d^{2} - 4}{7 d^{2} - 12 d - 4}} \frac{\frac{4 d^{2} + 7 d - 2}{35 d + 10}}{\frac{d^{2} - 4}{7 d^{2} - 12 d - 4}}$

121.

$\frac{10 m^{2} + 80 m}{3 m - 9} \cdot \frac{m^{2} + 4 m - 21}{m^{2} - 9 m + 20} \frac{10 m^{2} + 80 m}{3 m - 9} \cdot \frac{m^{2} + 4 m - 21}{m^{2} - 9 m + 20}$
$\div \frac{5 m^{2} + 10 m}{2 m - 10}$

122.

$\frac{4 n^{2} + 32 n}{3 n + 2} \cdot \frac{3 n^{2} - n - 2}{n^{2} + n - 30} \frac{4 n^{2} + 32 n}{3 n + 2} \cdot \frac{3 n^{2} - n - 2}{n^{2} + n - 30}$
$\div \frac{108 n^{2} - 24 n}{n + 6}$

123.

$\frac{12 p^{2} + 3 p}{p + 3} \div \frac{p^{2} + 2 p - 63}{p^{2} - p - 12} \frac{12 p^{2} + 3 p}{p + 3} \div \frac{p^{2} + 2 p - 63}{p^{2} - p - 12}$
$\cdot \frac{p - 7}{9 p^{3} - 9 p^{2}}$

124.

$\frac{6 q + 3}{9 q^{2} - 9 q} \div \frac{q^{2} + 14 q + 33}{q^{2} + 4 q - 5} \frac{6 q + 3}{9 q^{2} - 9 q} \div \frac{q^{2} + 14 q + 33}{q^{2} + 4 q - 5}$
$\cdot \frac{4 q^{2} + 12 q}{12 q + 6}$

Everyday Math

125.

Probability The director of large company is interviewing applicants for two identical jobs. If $w = w =$ the number of women applicants and $m = m =$ the number of men applicants, then the probability that two women are selected for the jobs is $\frac{w}{w + m} \cdot \frac{w - 1}{w + m - 1} . \frac{w}{w + m} \cdot \frac{w - 1}{w + m - 1} .$

ⓐ Simplify the probability by multiplying the two rational expressions.
ⓑ Find the probability that two women are selected when $w = 5 w = 5$ and $m = 10 m = 10$ .

126.

Area of a triangle The area of a triangle with base b and height h is $\frac{b h}{2} . \frac{b h}{2} .$ If the triangle is stretched to make a new triangle with base and height three times as much as in the original triangle, the area is $\frac{9 b h}{2} . \frac{9 b h}{2} .$ Calculate how the area of the new triangle compares to the area of the original triangle by dividing $\frac{9 b h}{2} \frac{9 b h}{2}$ by $\frac{b h}{2} \frac{b h}{2}$ .

Writing Exercises

127.

ⓐ Multiply $\frac{7}{4} \cdot \frac{9}{10} \frac{7}{4} \cdot \frac{9}{10}$ and explain all your steps.
ⓑ Multiply $\frac{n}{n - 3} \cdot \frac{9}{n + 3} \frac{n}{n - 3} \cdot \frac{9}{n + 3}$ and explain all your steps.
ⓒ Evaluate your answer to part (b) when $n = 7 . n = 7 .$ Did you get the same answer you got in part (a)? Why or why not?

128.

ⓐ Divide $\frac{24}{5} \div 6 \frac{24}{5} \div 6$ and explain all your steps.
ⓑ Divide $\frac{x^{2} - 1}{x} \div (x + 1) \frac{x^{2} - 1}{x} \div (x + 1)$ and explain all your steps.
ⓒ Evaluate your answer to part (b) when $x = 5 . x = 5 .$ Did you get the same answer you got in part (a)? Why or why not?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

The above image is a table with four columns and four rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads multiply rational expressions.”, the next row reads “divide rational expressions.”, the last row reads “after reviewing this checklist, what will you do to become confident for all objectives?” The remaining columns are blank.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?