Chapter 3, Section A2

Math Topics – Multiplying and Dividing Algebraic Fractions; Graphing Rational Equations

Multiplying Fractions

In a previous section, we looked at how multiplying by a fraction like means that you are taking half of something. For example, we know that half of 8 is 4. To show this using fractions, we multiply:

This is a lot of work to go through to get the answer we already know, 4! But the important part of this is that we can see that we multiply fractions straight across the tops (numerators) and straight across the bottoms (denominators).

Similarly, we know that half of a half gives us a quarter. For example, we can imagine a half of a dollar (50 cents), and then when we take a half of that, we get 25 cents, which is a quarter (of a dollar). The steps to get this answer also show multiplying straight across:

Thus, to multiply algebraic fractions, we follow the same rules, multiplying straight across the top and straight across the bottom.

Example 1 multiply Simplify if needed.

First, we multiply straight across by multiplying the numbers and adding the exponents: .

Next, we see if we can simplify anything. Note that both 12 and 10 can be divided by 2. That gives us .

We could also write 1.2 , since 12 / 10 = 1.2.

Either answer is correct, but sometimes you will see instructions to “write in terms of whole numbers only,” in which case, you should write instead of 1.2.

Example 2 multiply Simplify if needed.

Again, we first multiply straight across by multiplying the numbers and adding the exponents:

Now we simplify: 35 and 21 both have a factor of 7 in common. We divide both top and bottom (numerator and denominator) by 7 to get  .

Now look at the variables. Both numerator and denominator have y variables. Both numerator and denominator have z variables. To simplify, we have two different strategies: 1. Subtract the exponents, 2. Expand and “cancel”.

Let’s start with expand and cancel first. Even though “subtract the exponents” might seem faster, it can be harder to understand. And remember, understanding is more important than being fast.

Example 3 another multiply but with (x + 5)

then division

Then graphing

y = 1/x

y = 1/x – 2