In math, there are certain rules we learn once but use again and again throughout our math journey.
It’s a bit like learning to tie your shoes—once you master it, you don’t have to think about it every time you reach for the laces!
One such math rule is the associative property of multiplication.
Read on to find out what this rule means, see it in action with examples, test your knowledge in a quick quiz, and find answers to common questions.
Find Top-Rated K-12 Math Tutors Near You
The associative property might sound scary and complicated, but it doesn’t need to be.
So, what does it mean?
The associative property is a math rule that tells us we can group numbers in different ways without changing the answer.
Think about the word "associate"—it means to connect or group things together. In math, this means we can change how we group numbers when adding or multiplying, and the result will stay the same!
And why does this rule work for both addition and multiplication?
Since multiplication is repeated addition (for example, 3 × 4 is the same as 3 + 3 + 3 + 3), and addition follows this rule, multiplication does as well.
Let's see an example of the associative property in addition.
Say we have the expression 2 + 3 + 4
If we add the numbers from left to right, we will get:
(2 + 3) + 4
= 5 + 4
= 9
Now, let’s group the last two numbers in parentheses and see if the outcome changes:
(2 + 3) + 4
= 2 + 7
= 9
We see that the answer stays the same, no matter in which order we add the numbers.
The same is true for multiplication. Let's test it!
You May Also Like: What Is PEMDAS?
The associative property of multiplication tells us that when we multiply numbers, we can change how we group them without changing the result.
In math, we write it like so:
(a x b) x c = a x (b x c)
This rule also works for more than three numbers:
(a x b) x (c x d) = (a x d) x (b x c)
Let's try this in practice.
7 x 4 x 5
We can group the numbers in different ways, but the product stays the same:
(7 x 4) x 5 = 28 x 5 = 140
We can also use the associative property to rearrange numbers in a way that might be easier to solve mentally:
(7 x 4) x 5 = 7 x (4 x 5) = 140
No matter how we group the numbers, the final product remains 140!
Practice makes perfect!
Let's see how the associative property of multiplication works when a negative number is involved.
What is (–3) x 4 x 25?
We can start by grouping the first two numbers:
(–3 x 4) x 25 = –12 x 25 = –300
We can also change the way we group the numbers to make the problem easier:
(–3 x (4 x 25) = –3 x 100 = –300
However we choose to group the numbers, the answer stays the same: –300.
The associative property might seem simple, but it often raises some interesting questions. Let’s see the answers to a few of the ones we get often!
The associative property tells us that we can regroup numbers without changing the result:
(2 x 3) x 4 = 2 x (3 x 4) = 24
The commutative property tells us that the order of numbers does not change the result:
2 x 3 = 3 x 2 = 6
When working with basic multiplication, there are no exceptions—the associative property always applies.
Yes, the associative property applies even when we’re multiplying or adding with zero! For example:
(0 x 3) x 4 = 0 x (3 x 4) = 0
In case of multiplication, remember: No matter how you group the numbers, the result stays zero because anything multiplied by zero is zero.
Subtraction and division don’t follow the associative property because changing the grouping changes the result.
To show why the associative property doesn't work with subtraction, let's take 10 – 5 – 2 and group the numbers in two different ways:
(10 – 5) – 2 and 10 – (5 – 2)
We get two different results: 3 and 7.
The same goes for division. Let’s start with the expression 12 ÷ 6 ÷ 2 and group the numbers in two different ways:
(12 ÷ 6) ÷ 2 and 12 ÷ (6 ÷ 2)
Again, we get two different results: 1 and 4.
Take this quick quiz to test what you’ve learned about the associative property of multiplication.
Note: Some questions might have more than one correct answer.
Create your own user feedback survey
Mathnasium’s specially trained elementary school tutors work with students of all skill levels to help them master any math subject and topic, including the associative property of multiplication.
Discover our approach to elementary school tutoring:
At Mathnasium, we assess each student’s current skills and consider their unique academic needs to develop personalized learning plans that will put them on the best path toward math mastery.
Whether your student is looking to catch up, keep up, or get ahead in your math class, find a Mathnasium Learning Center near you, schedule an assessment, and enroll them today!
Answer a few questions to see how it works
My Child is:
Mathnasium meets your child where they are and helps them with the customized program they need, for any level of mathematics.
We have nearly 1,000 neighborhood centers nationwide. Get started now.
