If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Lesson 3: Associative property of multiplication

# Associative property of multiplication

Let's explore the associative property of multiplication! This video demonstrates that the order of multiplying numbers doesn't affect the result, using examples like 4 x 5 x 2. The associative property simplifies math.

## Want to join the conversation?

• Is there a difference between the associative and commutative laws? They seem to say the same thing--order does not matter.

The only difference I see is the parentheses in the associative law which is explicitly "associating" two numbers together.
• The commutative property lets you change the order of the numbers. This is the one that tells you that the order does not matter.
Example: 2 * (3 * 5) = (3 * 5) * 2
In this example: the "2" moved, but the parentheses still contain their same numbers.

The associative property tells use that we can regroup (move parentheses).
Example: 2 * (3 * 5) = (2 * 3) * 5
In this example, we regroup by moving the parentheses to now contain 2*3 rather than 3*5

The combination of these 2 properties lets us regroup and change the order of numbers being multiplied and we still get the same result.
• what is the direct simplification of the defenition of what the associative property is? I don't understand this definition.
• The associative property of multiplication let's us move / change the placement of grouping symbols. It does not move the numbers.
For example: (2 x 4) x 5 can be changed into 2 x (4 x 5)
Both expressions create the same result.
• This means that
(5 X 3) X 4 = (5 X 4) X 3 = (4 X 3) X 5
Is is correct ?
• (5x3) x4 = (5x4) x3 = (4x3) x5
The above statement is true.
• Does that mean that it could be written in different order and still be the same answer?
• YES that is exactly what it means!
• does anyone just love math after once hating it and saying that you would hate it for eternity
• I problably did. Without Sal, I would be hating it right now.
• Thank you this helped me so much
• I just do not get it. Can you make another video like that to explain it to me.
(1 vote)
• When doing mathematical operations, we put parentheses around the parts that need to get done first. For example, when we put parentheses around (4 × 5) in the following expression:

`2 × (4 × 5)`

...it just means we have to multiply the 4×5 first, before multiplying by 2. So we solve it as follows, by first multiplying 4×5:

`2 × (4 × 5) = 2 × 20`

...and then we can finish it off by multiplying by 2:

`2 × (4 × 5) = 2 × 20 = 40`

When they're talking about the "associative property of multiplication," all they really mean is that when you multiply things together, you can group them into parentheses any way you want, because the result will be the same. And because the part in parentheses gets done first, this means that you can use parentheses to do whichever part of the multiplication you want to do first. For example, all of these expressions give the same final result, even though the parentheses are in different places:

` 2 × 4 × 5 = 8 × 5 = 40`
`(2 × 4) × 5 = 8 × 5 = 40`
` 2 × (4 × 5) = 2 × 20 = 40`
• How do you use Associative property of Multiplication with 4 nummbers
• you do the same thing, like 3x2x9x7 = 378 7x2x9x3 = 378
• Does it need to be only odd or even?
• Every number that exists is either odd or even, so yes. You can mix odd and even numbers, or have two of the same. I hope this helped! Comment if you have any questions!
• until he had not told that they are not using the same numbers i had not noticed
(1 vote)

## Video transcript

- [Instructor] So, what we're gonna do is get a little bit of practicing multiple numbers together and we're gonna discover some things. So, first I want you to figure out what four times five times two is. Pause the video and try to figure it out on your own. Alright, so whatever your answer is, some of you might have done it this way, some of you might have said hey, what is four times five and then you multiplied it by two, so what you would really have done is you would have done four times five first, so that's why I put parentheses around that and then you would have multiplied by two and what would you have gotten? Weil, the four times five part, that is of course 20 and then you multiply that times two and you would get 40 which of course would be correct, four times five times two is indeed equal to 40. Now, what I want you to do now is as quickly as possible try to figure out what five times two times four is. Really quick, pause the video, try to figure that out. Well, some of you might have tried and you might have done it in a similar way where you tried to figure out five times two is first and you said okay, five times two is equal to 10 and then I'd multiply that times four and then you would say well, gee, this is same thing as I got last time. Is there something interesting going on? And the interesting thing that you might realize is in both cases we're multiplying the same three numbers. We are just doing it in a different order. Here we multiplied four times, we wrote it out in a different order, four times five times two. Here we wrote five times two times four. Here we did the four times five first, here we did the five times two first but notice we got the same result. Now, I'd encourage you, pause this video. Try to multiply these numbers in any order, maybe you do two times four first. In fact, let's just do that. Let's do two times four, two times four and then multiply that by five. What is this going to be equal to? Well, you might notice again this is two times four is eight, you multiply that times five. Well, once again, we got 40, so you might see a pattern here. It doesn't matter which order we multiply these things in. In fact, you could write four times five times two. You could do the four times five first, four times fives times two or you could do four times five times two, so you could do four times five times two. So, it doesn't matter which order you multiply these things in. In every case you are going to get 40. Now, there's a very fancy term for this, the associative property of multiplication but the main realization is and it's not just true with the three numbers, in fact, you've seen something similar with two numbers where it doesn't matter what order you multiply them in but what you see with three numbers and even if you tried it with four or five or really 1,000 numbers being multiplied together, as long as you're just multiplying them all, it doesn't matter what order you're doing it with. It doesn't matter in what order you associate them with. Here we did four times five first, four times five first, here we did five times two first but in either case we got the same result and I'd encourage you, after this video, try to draw it out. Try to think about why that actually makes intuitive sense, why this is true in the world and it's nice because it simplifies our life when we're doing mathematics and not only now but in our future mathematical career.