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Associative Property of Multiplication

Omkar Phatak Oct 17, 2018
If you are curious about what exactly is the associative property of multiplication, then this post will be an interesting read. Along with illustrative examples, you will find it lucidly explained here.
There are two sets of mathematical concepts which you will study. They include numbers or objects and operators. Every mathematical operation is a symbolic manipulation of abstract objects by certain operators.
The most basic of objects are natural and whole numbers, while the most fundamental mathematical operations are division, multiplication, addition, and subtraction. Their corresponding operators (/, x, +, - ) are applied to numbers to form mathematical expressions.
A linking of such expressions, by the 'equal to (=)' relation creates an equation in mathematics. Each of the basic operations are described by certain properties, which define their usage in various circumstances.
Consider the expression (2 x 2), which is 2 multiplied by 2. An equivalent way of saying the same thing would be '2 added 2 times'. So (2 x 2) is in a way, (2 + 2), which makes the result to be '4'. Thus, the product of 2 multiplied by 2 is 4. Thus, one can see that multiplication is actually addition.


The associative property deals with the way numbers are grouped together in brackets, when being multiplied. Here is the property, stated in its simplest form:

"Irrespective of how numbers being multiplied are grouped together, the end product of multiplication remains the same."
Such properties of mathematical operations are understood better, when presented in the form of an equation:

"a x (b x c) = (a x b) x c"
Here a, b, and c are variables or numbers being multiplied with each other. As a rule, in arithmetic, the expression in the bracket is always solved first and the rest of the expressions are simplified later.
So in this equation, even when (b x c) expression is calculated first or when (a x b) expression is calculated first, the end result remains the same. This may seem trivial, but it is necessary that this property be stated for completeness.


Actual worked examples make it easier for any student of mathematics to understand any concept deeply. Here are some examples which demonstrate the property.
  • 24 x (2 x 3) = (24 x 2) x 3
  • (10 x 8) x 9 = 10 x (8 x 9)
  • m x n x (p x q) = m x (n x p) x q
  • 25 x (2 x 1) = (25 x 2) x 1
These examples actually demonstrate how you are free to group terms, according to your convenience, while multiplying numbers. This is the most subtle of all multiplication properties, which allows for easy grouping of multiplicative terms, making it easier for you to calculate the end result.