Multiplying And Dividing Rational Expressions

In this video, we are going to multiply and divide rational expressions. After you finish this lesson, view all of our Algebra 1 lessons and practice problems.

We can start by either directly performing the operation or reduce across.

For example:

First, reduce the numbers that are across from each other

Further reduce, which can be done vertically with the 8 and the 4

Multiply

And we have

With variables, it is the same concept

Factor the expressions if necessary

Reduce diagonally

Multiple across

Division is similar to multiplication.

Remember to Keep, Change, Flip

For example:

Keep the first fraction, change the operation from division to multiplication, and flip the second fraction

Factor if necessary

Reduce expressions

Multiply across

Examples of Multiplying And Dividing Rational Expressions

Example 1

Solve
First, multiply the numerators together, and multiply the denominators together

Simplify by finding common factors in the numerator and denominator
So, cancel out and

Therefore, the final answer is

Example 2

First, lets factor out

Now, in order to solve this, we do keep-change-flip.
Keep the first one. Then change to multiplication. And lastly, flip the second expression.

Then, simplify by finding common factors in the numerator and denominator
So, cancel out

Now, is cancelled. Then and
The final answer is

Video-Lesson Transcript

Let’s go over how to multiply and divide rational expressions.

For example:

Just like normal fractions, we can multiply straight across or cross-cancel first.

I advise you to do cross-cancel first.

Just a review,

We’ll cross-cancel by finding the greatest common factor of and – which is .

Then identify what number goes into and . So, does.

Then, we can reduce the second expression and we’ll have

Let’s multiply across

Let’s do the same thing with our example.

Let’s see what we can reduce.

and can be reduced.

Then let’s subtract the exponents of .

Now, we can multiply across.

Let’s have a more complex example.

Here, we can’t cross-cancel because the terms are being added and subtracted.

Because in cross-cancel, things are being multiplied.

An example of this is:

This can be cross-cancelled because everything is multiplied.

What if terms aren’t multiplied?

We have to factor them out first. So we can come up with multiplication.

Only then can we cross-cancel.

Now that we factored this, let’s see what we can reduce.

We can cancel and

So, our final answer is

Now, let’s look at division.

Before we go into the lesson, let’s have a quick look at the division of fractions.

In order to solve this, we do keep-change-flip.

Keep the first one. Then change to multiplication. And lastly, flip the second expression.

Then follow the same rule in multiplication.

Cross-cancel if you can.

Now, let’s apply the same rules to this problem.

So let’s do keep-change-flip.

Keep the first one, change to multiplication and flip the last one.

Let’s try to factor this before we reduce.

Now, let’s reduce.

cancelled. Then and . Then we’ll subtract the exponent of the -es.

To get the final answer of