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# Distributive property over addition

Video transcript

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.