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 performs the operation or reduce across.

For example: \frac{15}{4}\times\frac{32}{20}

First, reduce the numbers that are across from each other
\frac{15}{4}\times\frac{32}{20}
\frac{3}{1}\times\frac{8}{4}

Further reduce, which can be done vertically with the 8 and the 4
\frac{3}{1}\times\frac{2}{1}

Multiply
\frac{6}{1}

And we have
6

With variables, it is the same concept
\frac{3x+15}{x^2+7x+12}\times\frac{x^2-9}{x+5}

Factor the expressions if necessary
\frac{3(x+5)}{(x+4)(x+3)}\times\frac{(x+3)(x+3)}{(x+5)}

Reduce diagonally
\frac{3}{(x+4)}\times\frac{(x+3)}{1}

Multiple across
\frac{3(x+3)}{x+4}

Division is similar to multiplication.
Remember to Keep, Change, Flip

For example: \frac{x+8}{3x}\div\frac{4x+32}{15x^4}

Keep the first fraction, change the operation from division to multiplication, and flip the second fraction
\frac{x+8}{3x}\times\frac{15x^4}{4x+32}

Factor if necessary
\frac{x+8}{3x}\times\frac{15x^4}{4(x+8)}

Reduce expressions
\frac{1}{1}\times\frac{5x^3}{4}

Multiply across
\frac{5x^3}{4}

Examples of Multiplying And Dividing Rational Expressions

Example 1

Solve \dfrac{5a^2}{14}\cdot\dfrac{7}{10a^3}
First, multiply the numerators together, and multiply the denominators together
\dfrac{35a^2}{140a^3}
Simplify by finding common factors in the numerator and denominator
So, cancel out 35 and a^2
\dfrac{35 \cdot a^2}{4 \cdot 35 \cdot a^2\cdot a}
Therefore, the final answer is
\dfrac{1}{4a}

Example 2

\dfrac{3x^2}{x+2}\div\dfrac{6x^4}{x^2+5x+6}
First, lets factor out x^2+5x+6

\dfrac{3x^2}{x+2}\div\dfrac{6x^4}{(x+3)(x+2)}
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.
\dfrac{3x^2}{x+2}\times\dfrac{(x+3)(x+2)}{6x^4}
Then, simplify by finding common factors in the numerator and denominator
So, cancel out x+2
{3x^2}\times\dfrac{(x+3)}{6x^4}
Now, x^2 is cancelled. Then 3 and 6
The final answer is
\dfrac{x+3}{3x^2}

Video-Lesson Transcript

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

For example:

\dfrac{3x^2}{4} \times \dfrac{12y}{x^3}

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

I advise you to do cross-cancel first.

Just a review,

\dfrac{15}{4} \times \dfrac{32}{20}

We’ll cross-cancel by finding the greatest common factor of 4 and 8 – which is 4.

Then identify what number goes into 15 and 20. So, 5 does.

= \dfrac{3}{1} \times \dfrac{8}{4}

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

= \dfrac{3}{1} \times \dfrac{2}{1}

Let’s multiply across

= \dfrac{6}{1}
= 6

Let’s do the same thing with our example.

\dfrac{3x^2}{4} \times \dfrac{12y}{x^3}

Let’s see what we can reduce.

4 and 12 can be reduced.

Then let’s subtract the exponents of x.

= \dfrac{3}{1} \times \dfrac{3y}{x^1}

Now, we can multiply across.

= \dfrac{9y}{x}

Multiplying And Dividing Rational Expressions

Let’s have a more complex example.

\dfrac{3x + 15}{x^2 + 7x + 12} \times \dfrac{x^2 - 9}{x + 5}

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:

\dfrac{(3) (4)}{(5)} \times \dfrac{(10) (7)}{(6)}

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.

\dfrac{3 (x + 5)}{(x + 4) (x + 3)} \times \dfrac{(x + 3) (x - 3)}{(x + 5)}

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

We can cancel x + 5 and x +3

So, our final answer is

\dfrac{3 (x - 3)}{x + 4}

Now, let’s look at division.

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

\dfrac{12}{5} \div \dfrac{2}{9}

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

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

\dfrac{12}{5} \times \dfrac{9}{2}

Then follow the same rule in multiplication.

Cross-cancel if you can.

= \dfrac{6}{5} \times \dfrac{9}{1}
= \dfrac{54}{5}

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

\dfrac{x + 8}{3x} \div \dfrac{4x + 32}{15x^4}

So let’s do keep-change-flip.

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

= \dfrac{x + 8}{3x} \times \dfrac{15x^4}{4x + 32}

Let’s try to factor this before we reduce.

= \dfrac{x + 8}{3x} \times \dfrac{15x^4}{4 (x + 8)}

Now, let’s reduce.

x +8 cancelled. Then 3 and 15. Then we’ll subtract the exponent of the x-es.

To get the final answer of

= \dfrac{5x^3}{4}