Simplifying Using the Distributive Property

Date: 09/07/99 at 17:57:42
From: Christina
Subject: Math - Distributive Property

How do you actually work out this kind of problem?

Directions: Simplify each expression.

     12(c+3d+4e)+2(2c+d+6e)

This is really confusing, when I have those big math problems. This 
one is kind of hard too:

     9ab+8ab-7ab

Thank you very much.

Date: 09/12/1999 at 21:12:21
From: Doctor Jody
Subject: Re: Math - Distributive Property

Hi Christina,

I can see how it can be confusing. There's a lot to do. Let's try to 
break it down into simpler tasks - then it might seem less confusing.

The first thing I would do is put some space between the different 
parts of the expression:

     12(c + 3d + 4e) + 2(2c + d + 6e)

It seems less cluttered to me when I write it like this, and that 
helps unclutter my thinking.

It looks as if we have some multiplying and some adding to do. You 
have to do multiplication in two parts of the problem, and then add 
those results together. Let's look at just one part of the problem 
where you have to do multiplication:

     12(c + 3d + 4e)

We can't add the items in the parentheses to each other. Do you know 
why? Here, the only thing I can think of to do is multiply the 12 by 
each of the items inside the parentheses (applying the distributive 
property over addition.)

After multiplying 12 by each of the items in the parentheses, we get:

     12c + 36d + 48e

We can't add these items together, either. So it's as simple as it can 
get. Now it's time to look at the other part of the problem:

     2(2c + d + 6e)

It looks as if we need to do the same thing to it, for the same 
reasons, so:

     4c + 2d + 12e

Do you see how I did that? Looking back at the original problem, it 
shows that these two parts of the problem need to be added together. 
Let's put them side by side again, with an addition sign between them, 
to make it clear:

     12c + 36d + 48e   +   4c + 2d + 12e

Because they are all plus signs, it doesn't matter if there are 
parentheses, and we can add them in any order we like, as long as it 
makes sense. In this case, the thing to do is add the terms that 
share the same variable, also called "like terms." So,

     12c +  4c = 16c
     36d +  2d = 38d
     48e + 12e = 60e

Giving us:

     16c + 38d + 60e

Can you simplify this expression any more?

Try the second problem that you wrote above, doing it the same way as 
I did the last part for the first problem. You will probably see what 
to do. Here's a nice explanation of the distributive property from the 
Dr. Math archives, if you need more help with it:

  What is the Distributive Property?
  http://mathforum.org/dr.math/problems/distribute.html   

Write back if you have any more questions, and please try to explain 
exactly where your problem is. I can help you much more that way.

- Doctor Jody, The Math Forum
  http://mathforum.org/dr.math/