8.4 Distributive Laws  
Distributive laws connect multiplication with addition and subtraction. We usually call these “multiplying in” or “breaking out”.

We begin with an example:


You know that 3 · 9 = 27

We can divide 9 into two terms, for example (5 + 4).

If we multiply 3 by (5 + 4), will we get the same answer as if we had multiplied 3 · 9? What do you think?

We can illustrate this in pictures:
Here you see that we get the same result, 27.

You can try to divide 9 into other terms and multiply by 3. Check and see if you get the same results.

In you mathematics book, it may be explained like this:

3(5 + 4) = (3 5) + (3 4) = 15 + 12 = 27

When you calculated with written multiplication, you might use the distributive laws without being aware of it. 

Here is an example when you use it:
2 2
· 5 1
1 1 0
5 · 22 = 5 · (2 + 20) =
= 5 · 2 + 5 · 20 =
= 10 + 100 = 110

Now you have learned to “multiply in”, and now we can show you how to “break out”.

15 + 12 has a common factor, 3.

15 can be written (5 · 3) and 12 can be written (4 · 3).  We can then write the addition as follows:
15
+
12 = 27
(5 3)
+
(4 3) = 27
We break out the common factor 3:

You have learned that the distributive law applies for multiplication with addition in parenthesis. Does it work even for multiplication with subtraction in parenthesis?  We will look at an example:

5
(8 - 6)

Multiply in:
5 · (8 - 6) = (5
8) - (5
6) = 40 - 30 = 10

Calculate the parenthesis first:
5 · (8 - 6) = 5
2 = 10

We get the same answer and the rule seems to apply no matter if it´s multiplication with addition or subtraction in the parenthesis.


We can summarise the calculation rule for multiplication with addition or subtraction in the parenthesis in something called the distributive law:
and


If you want to learn more about breaking out you can read more about it in chapter 5.1 in the section Algebra.