The two most commonly used operations in math are addition and multiplication.

And although we intermingle addition with multiplication (along with division and subtraction), these two can *only* mix using the distributive property.

### How the distributive property works

Multiplication, in a crude sense, is a shortcut of addition. Like 3 x 8 is the number 8 added to itself three times. Or 8 + 8 + 8.

Look at 3 x (6 + 2). This is the number (6 + 2) added to itself three times. (Refrain from adding those two together, no matter how tempting. Leave them separate as 6 + 2.)

So we have (6 + 2) + (6 + 2) + (6 + 2).

And we know that when it comes to only adding, we don’t need any parenthesis. Which gives us

6 + 2 + 6 + 2 + 6 + 2

Also, addition is *commutative*. So we can scramble these numbers around.

6 + 6 + 6 + 2 + 2 + 2

And if we want, we can squish them together in parenthesis (seems lame, but stay with me…)

(6 + 6 + 6) + (2 + 2 + 2)

And, hey! We know that 6 + 6 + 6 means 3 x 6 (and the same with 2’s) so we have:

(3 x 6) + (3 x 2)

Remember what we started with. Put it together with what we ended with. And we’ve just demonstrated (the long way) that

3 x (6 + 2) = (3 x 6) + (3 x 2)

or, more visually…

**3 x (6 + 2) = (3 x 6) + (3 x 2)**

This is the ** Distributive Property of Multiplication Over Addition**. (That’s the fancy name for the distributive property.)

### It works with something else, too…

Since exponents are a shortcut of multiplication, we can play the same game.

Consider (6 x 2)^{3}

This is (6 x 2) x (6 x 2) x (6 x 2)

And we can get rid of parenthesis and scramble:

6 x 6 x 6 x 2 x 2 x 2

And re-group:

6^{3} x 2^{3}

Showing that:

(6 x 2)^{3} = 6^{3} x 2^{3}

This, by the way, is the **Distributive Property of Exponentiation Over Multiplication!**

### Yes – this IS useful.

Seems ridiculous, right?

But when learning algebra, students often get lost when faced with 3(*y* + 2). “What do I do now?” they think, going into panic mode.

If they know that *y* is just a number with lipstick on (as a disguise), they can think about the above example.

Remembering what it means, is the key to knowing what to do with the algebra stuff!

###### You might also like:

- Distributive Property Gets Destroyed by the Lattice Method
- The Order of Operations Explained: Parenthesis
- How Calculators Inhibit Learning the Distributive Property in Algebra
- Practice, Practice, Practice – Really?

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