7.2 Multiplying rational expressions

Multiplying fractions is easy: you just multiply the tops, and multiply the bottoms. For instance,

Now, you may notice that $\frac{\text{42}}{\text{77}}$ can be simplified, since 7 goes into the top and bottom. $\frac{\text{42}}{\text{77}}=\frac{\text{42}÷7}{\text{77}÷7}=\frac{6}{\text{11}}$ . So $\frac{\text{42}}{\text{77}}$ is the correct answer, but $\frac{6}{\text{11}}$ is also the correct answer (since they are the same number), and it’s a good bit simpler.

In fact, we could have jumped straight to the simplest answer first, and avoided dealing with all those big numbers, if we had noticed that we have a 7 in the numerator and a 7 in the denominator, and cancelled them before we even multiplied!

This is a great time-saver, and you’re also a lot less likely to make mistakes.

When multiplying fractions...

But it’s critical to remember that this rule only applies when you are multiplying fractions : not when you are adding, subtracting, or dividing.

As you might guess, all this review of basic fractions is useful because, once again, rational expressions work the same way.

Multiplying rational expressions

$\frac{{\mathrm{3x}}^2-\text{21}x-\text{24}}{x^2-\text{16}}\cdot \frac{x^2-\mathrm{6x}+8}{\mathrm{3x}+3}$	The problem
$=\frac{3(x-8)(x+1)}{(x+4)(x-4)}\cdot \frac{(x-2)(x-4)}{3(x+1)}$	Always begin rational expression problems by factoring! Note that for the first element you begin by factoring out the common 3, and then factoring the remaining expression.
	When multiplying fractions, you can cancel anything on top with anything on the bottom, even across different fractions
$=\frac{(x-8)(x-2)}{x+4}$	Now, just see what you’re left with. Note that you could rewrite the top as $x^2-\text{10}x+\text{16}$ but it’s generally easier to work with in factored form.

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Dividing rational expressions

To divide fractions, you flip the bottom one, and then multiply.

After the “flipping” stage, all the considerations are exactly the same as multiplying.