Multiplying and Dividing Rational Expressions
Multiplying rational expressions is similar to multiplying fractions in
arithmetic.
Procedure
To Multiply Rational Expressions
Step 1 Factor the numerators and denominators.
Step 2 Cancel all pairs of factors common to the numerators and
denominators.
Step 3 Multiply the numerators, and then multiply the denominators.
Example 1
Simplify:
![](./articles_imgs/705/multip64.gif)
Solution
Step 1 Factor.
Step 2 Cancel common factors.
Step 3 Multiply.
|
![](./articles_imgs/705/multip64.gif)
![](./articles_imgs/705/multip65.gif)
![](./articles_imgs/705/multip66.gif)
![](./articles_imgs/705/multip67.gif) |
The result is .
Note
In your final answer, you may leave the
parentheses. Or, you may use the
Distributive Property to remove them. So,
may be written
as
.
Dividing rational expressions is similar to dividing fractions in arithmetic.
Procedure
To Divide Rational Expressions
Step 1 Invert the second rational expression and change
÷ to ·.
Step 2 Multiply.
Example 2
Simplify:
![](./articles_imgs/705/multip70.gif)
Solution
Step 1 Invert the second rational
expression and change ÷ to
·.
Step 2 Multiply.
Factor.
Cancel.
Simplify. |
![](./articles_imgs/705/multip70.gif)
![](./articles_imgs/705/multip71.gif)
![](./articles_imgs/705/multip72.gif)
![](./articles_imgs/705/multip73.gif)
|
Thus, the result is .
Note:
can be written as
![](./articles_imgs/705/multip76.gif)
Some rational expressions contain both multiplication and division. To
simplify these expressions, we can begin by changing division to
multiplication.
Example 3
Simplify:
![](./articles_imgs/705/multip77.gif)
Solution
First, convert the division to
multiplication.
Factor.
Cancel pairs of common factors.
Multiply.
|
![](./articles_imgs/705/multip77.gif)
![](./articles_imgs/705/multip78.gif)
![](./articles_imgs/705/multip79.gif)
![](./articles_imgs/705/multip80.gif)
![](./articles_imgs/705/multip81.gif) |
The result is
.
|