# Rational Expressions and Equations

## Multiply and Divide Rational Expressions


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

### 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  is . To find the reciprocal we simply put the numerator in the denominator and the denominator in the numerator. We “flip” the fraction.
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.
Before doing the next example, let’s look at how we divide a fraction by a whole number. When we divide , we first write 4 as a fraction so that we can find its reciprocal.
We do the same thing when we divide rational expressions.
Remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.
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.

### Key Concepts

1. Multiplication of Rational Expressions


2. Multiply a Rational Expression


3. Division of Rational Expressions


4. Divide Rational Expressions



### Practice Makes Perfect

Multiply Rational Expressions
In the following exercises, multiply.
Divide Rational Expressions
In the following exercises, divide.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?