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Distributive property over addition

Learn how to apply the distributive law of multiplication over addition and why it works. This is sometimes just called the distributive law or the distributive property. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • male robot hal style avatar for user Ryuudo
    I remember using this in Algebra but why were we forced to use this law to calculate instead of using the traditional way of solving whats in the parentheses first, since both ways gives the same answer.
    (432 votes)
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    • female robot grace style avatar for user Jade Lauron
      This is preparation for later, when you might have variables instead of numbers. Sure 4(8+3) is needlessly complex when written as (4*8)+(4*3)=44 but soon it will be 4(8+x)=44 and you'll have to solve for x.

      At that point, it is easier to go:
      (4*8)+(4x) =44
      32 + 4x =44
      4x =12
      x = 3

      Working with numbers first helps you to understand how the above solution works. Hope this helps.
      (447 votes)
  • duskpin ultimate style avatar for user Josephine Perry
    Okay, so I understand the distributive property just fine but when I went to take the practice for it, it wanted me to find the greatest common factor and none of the videos talked about HOW to find the greatest common factor. Can any one help me out?
    (14 votes)
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  • mr pants orange style avatar for user Jovanny528
    one question i had when he said 4times(8+3) but the equation is actually like 4(8+3) and i don't get how are you supposed to know if there's a times table there.its on 19-39 on video.
    (13 votes)
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    • piceratops ultimate style avatar for user Hecretary Bird
      If there is no space between two different quantities, it is our convention that those quantities are multiplied together. Doing this will make it easier to visualize algebra, as you start separating expressions into terms unconsciously.
      4 (8 + 3) is the same as (8 + 3) * 4, which is 44.
      (7 votes)
  • spunky sam orange style avatar for user Jack FrancisV
    i need help to this problem 2(5-v)
    (10 votes)
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  • duskpin sapling style avatar for user keiaja.1334699
    you confusing me with this?
    (11 votes)
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  • duskpin tree style avatar for user Annabeth~
    why is the distributive property important in math? how can it help you? isn't just doing 4x(8+3) easier than breaking it up and do 4x8+4x3?
    (8 votes)
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  • starky ultimate style avatar for user IMJST
    i dont understand how it works but i can do it
    (4 votes)
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    • duskpin ultimate style avatar for user Elaina163
      Ok so what this section is trying to say is this equation 4(2+4r) is the same as this equation 8+16r. The reason why they are the same is because in the parentheses you add them together right? That would make a total of those two numbers. Those two numbers are then multiplied by the number outside the parentheses. So in doing so it would mean the same if you would multiply them all by the same number first. You would get the same answer, and it would be helpful for different occasions! This is a choppy reply that barely makes sense so you can always make a simpler and better explanation. Hope this helps!
      (9 votes)
  • blobby green style avatar for user thomasgamble
    i need the anser to this problem 7(-9r-5) pls help
    (4 votes)
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  • stelly blue style avatar for user cherie.ashe
    I"m a master at algeba right?
    (3 votes)
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  • starky ultimate style avatar for user ͢∘⊽∘
    Help me with the distributive property. Please? If you add numbers to add other numbers, isn't that the communitiave property? I can't get it. It's so confusing for me, and I want to scream sometimes....In a problem at school, it really "tugged" at me, and I couldn't get it! May y'all help me? PLEASE! PLEASĖ? PLEAS-!"
    (3 votes)
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    • hopper cool style avatar for user Philip
      The commutative property means when the order of the values switched (still using the same operations) then the same result will be obtained. For example, 1+2=3 while 2+1=3 as well. 2*5=10 while 5*2=10 as well.

      The literal definition of the distributive property is that multiplying a value by its sum or difference, you will get the same result.

      Let's take 7*6 for an example, which equals 42.
      If we split the 6 into two values, one added by another, we can get 7(2+4). 7*2=14, and 7*4=28. 14+28=42

      There is of course more to why this works than of what I am showing, but the main thing is this: multiplication is repeated addition. You can think of 7*6 as adding 7 six times (7+7+7+7+7+7). Having 7(2+4) is just a different way to express it: we are adding 7 six times, except we first add the 7 two times, then add the 7 four times for a total of six 7s.

      With variables, the distributive property provides an extra method in rewriting some annoying expressions, especially when more than 1 variable may be involved.

      For example, if we have b*(c+d). c and d are not equal so we cannot combine them (in ways of adding like-variables and placing a coefficient to represent "how many times the variable was added". However, the distributive property lets us change b*(c+d) into bc+bd. Even if we do not really know the values of the variables, the notion is that c is being added by d, but you "add c b times more than before", and "add d b times more than before".
      Experiment with different values (but make sure whatever are marked as a same variable are equal values).
      (6 votes)

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.