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. 