Associative Property. The associative property states that when adding or multiplying a series of numbers, it does not matter how the terms are ordered. Remember that simplifying within the parenthesis is the first step using the order of operations.
The rule for the associative property. The numbers are grouped or associated with one another. Let’s look at a couple of examples.
(a+b)+c = a+(b+c)
So, again, we’re going to solve what’s in the parenthesis first. Let’s try this with:
(2+3)+4 = 2+(3+4)
We’re going to follow our order of operations or PEMDAS. We start by adding 2+3 = 5, 5 plus 4 equals 9. 3+4 equals 7, plus 2 equals 9.
5+4 is 9, and that does equal 2+7, which is also 9.
Since we’re doing addition, it doesn’t matter what we do first.
Let’s see what happens when we do the multiplication side. Again, I’m going to use:
2*(3*4) = (2*3)*4
According to PEMDAS, we have to start inside our parenthesis, so that’s 2 times, 3 times 4 is 12, is equal to, again, starting at our parenthesis, 2 times 3 is 6, times 4.
2 times 12 is 24, and that does equal 6 times 4, which is also 24.
Like the commutative property, the associative property is only true for addition and multiplication. Let’s look at what happens when we try to use it on subtraction and division.
If we try to use the associative property, it would look like (a-b)-c, but again, it’s not going to be equal to a-(b-c).
So, I’m going to use numbers like (8-4)-2 does not equal 8-(4-2).
Again, using order of operations, 8 minus 4 is 4, minus 2 does not equal 8 minus 4 minus 2 is 2.
4 minus 2 is 2, and that does not equal 8 minus 2, which is 6.
As you can see, the associative property does not work on subtraction. Let’s use it on division now.
So, with division, it would look like (a÷b)÷c, again, will not equal a÷(b÷c).
I’m going to use the same numbers we used for subtraction. So, (8÷4)÷2 does not equal 8÷(4÷2).
Let’s simplify. Order of operations, 8 divided by 4 must be done first, 2 divided by 2, does not equal, again, parenthesis first, 8 divided by 4 divided by 2, is 2.
2 divided by 2 is 1, and that does not equal 8 divided by 2, which is 4.
And that’s the associative property. The next lesson: Commutative Property, both lessons are included in our Practice Tests.