The distributive property
The Distributive Property The Distributive Property
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- Rewrite the expression 4 times, and then in parentheses
- we have 8 plus 3, using the distributive law of
- multiplication over addition.
- Then simplify the expression.
- So let's just try to solve this or evaluate this
- expression, then we'll talk a little bit about the
- distributive law of multiplication over addition,
- usually just called the distributive law.
- So we have 4 times 8 plus 8 plus 3.
- Now there's two ways to do it.
- Normally, when you have parentheses, your inclination
- is, well, let me just evaluate what's in the parentheses
- first and then worry about what's outside of the
- parentheses, and we can do that fairly easily here.
- We can evaluate what 8 plus 3 is.
- 8 plus 3 is 11.
- So if we do that-- let me do that in this direction.
- So if we do that, we get 4 times, and in parentheses we
- have an 11.
- 8 plus 3 is 11, and then this is going to be equal to--
- well, 4 times 11 is just 44, so you can
- evaluate it that way.
- But they want us to use the distributive law of
- multiplication.
- We did not use the distributive law just now.
- We just evaluated the expression.
- We used the parentheses first, then multiplied by 4.
- In the distributive law, we multiply by 4 first. And it's
- called the distributive law because you distribute the 4,
- and we're going to think about what that means.
- So in the distributive law, what this will become, it'll
- become 4 times 8 plus 4 times 3, and we're going to think
- about why that is in a second.
- So this is going to be equal to 4 times 8 plus 4 times 3.
- A lot of people's first instinct is just to multiply
- the 4 times the 8, but no!
- You have to distribute the 4.
- You have to multiply it times the 8 and times the 3.
- This is right here.
- This is the distributive property in action right here.
- Distributive property in action.
- And then when you evaluate it-- and I'm going to show you
- in kind of a visual way why this works.
- But then when you evaluate it, 4 times 8-- I'll do this in a
- different color-- 4 times 8 is 32, and then so we have 32
- plus 4 times 3.
- 4 times 3 is 12 and 32 plus 12 is equal to 44.
- That is also equal to 44, so you can get it either way.
- But when they want us to use the distributive law, you'd
- distribute the 4 first. Now let's think
- about why that happens.
- Let's visualize just what 8 plus 3 is.
- Let me draw eight of something.
- So one, two, three, four, five, six,
- seven, eight, right?
- And then we're going to add to that three of something, of
- maybe the same thing.
- One, two, three.
- So you can imagine this is what we have inside of the
- parentheses.
- We have 8 circles plus 3 circles.
- Now, when we're multiplying this whole thing, this whole
- thing times 4, what does that mean?
- Well, that means we're just going to add this to itself
- four times.
- Let me do that with a copy and paste.
- Copy and paste.
- Let me copy and then let me paste.
- There you go.
- That's two.
- That's one, two, three, and then we have four, and we're
- going to add them all together.
- So this is literally what?
- Four times, right?
- Let me go back to the drawing tool.
- We have it one, two, three, four times this expression,
- which is 8 plus 3.
- Now, what is this thing over here?
- If you were to count all of this stuff, you would get 44.
- But what is this thing over here?
- Well, that's 8 added to itself four times.
- You could imagine you're adding all of these.
- So what's 8 added to itself four times?
- That is 4 times 8.
- So this is 4 times 8, and what is this
- over here in the orange?
- We have one, two, three, four times.
- Well, each time we have three.
- So it's 4 times this right here.
- This right here is 4 times 3.
- So you see why the distributive property works.
- If you do 4 times 8 plus 3, you have to multiply-- when
- you, I guess you could imagine, duplicate the thing
- four times, both the 8 and the 3 is getting duplicated four
- times or it's being added to itself four times, and that's
- why we distribute the 4.
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