Raakhee Lakhan

Multiplication is one of the most basic elementary arithmetic operations that a student learns about while growing up. In elementary mathematics, multiplication is a more advanced way of adding a number multiple times. The very foundation of multiplication lies with the concept of repeated addition and hence the operation of multiplication follows the same properties as the addition operator. One of these properties is the associative property of multiplication.

To “associate” means to connect or join with something. The associative property of multiplication says that while multiplying three numbers, regardless of the way the numbers are grouped, the end result will always be the same.Let’s try to understand the associative property of multiplication with an example:

Let’s try multiplying the numbers 2, 3, and 5.

Now, we can multiply these numbers in different ways.

We could first multiply 2 and 3 and then multiply their product with 5.

Or we could multiply 3 and 5 first and then multiply the product of these two numbers with

2.

As we can see, the product in both the cases is the same. This property, where the order in which the three numbers are multiplied does not affect the result, is called the associative property of multiplication.

Since addition is the foundation of multiplication, the associative property is only followed by addition and multiplication. The law of associativity does not apply to the operations of subtraction and division.

Example 1: Solve the expression $6 \times 7 \times 8$ in two different ways.

Solution:

Grouping the first two terms in the expression,

$(6 \times 7) \times 8

$= (42) \times 8$

$= 336$

Grouping the second two terms in the expression,

$6 \times (7 \times 8)$

$= 6 \times (56)$

$= 336$

Example 2: Does the given equation show the associative property of multiplication?

$2 \times 3 \times 4 = 3 \times 2 \times 4$

Solution: For an equation to show the associative property of multiplication, a minimum of three numbers must be multiplied. The given equation is the multiplication of 3, 2, and 4. The order of the numbers reversed gives the same answer, that is, 24. Thus, it shows an associative property.

Example 3: Use associative property of multiplication to find a and b in the equation,

$(3 \times a) \times 9 = 3 \times (4 \times b)$Solution: If the equation follows the associative property of multiplication, though grouped differently, the three terms on either side of the equation should be the same. 3 is present on either side. It follows that $a = 4$ and $9 = 6$.

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