What is associative property in multiplication?

Mathematics also provides us with certain manipulative principles to help with this problem. We can perform different arithmetic operations using them and solve complex equations with ease. The three main properties that form the backbone of math are:

In the following article, we’ll take a closer look at the associative property of multiplication.

Before we answer the question of what is the associative property of multiplication, let us first understand what associative means and some other elementary concepts.

To “associate” means to connect or join with something. In the context of mathematical operations, this means that the way numbers are grouped under a mathematical operation does not change the result. By grouping, we mean how the brackets are placed in the given algebraic expression. In simpler terms, the mathematical operation result remains the same irrespective of how the numbers are grouped.

Now that we’ve learnt a bit about grouping, let us answer the question of what is the associative property of multiplication in detail.

As per the associative property of multiplication definition, if three or more terms are multiplied together, we obtain the same end answer irrespective of how the terms are grouped.

Associative operations are abundant in mathematics; in fact, many algebraic structures (such as semigroups and categories) explicitly require their binary operations to be associative.

However, many significant and intriguing mathematical operations are non-associative. A few examples of this include subtraction, division, exponentiation, and vector cross product. We shall see these operations in detail a little later on. A real-world example of this is the addition of floating-point numbers in computer science which is not associative. The selection of how we associate an expression has a significant impact on the rounding error.

Although the associative property of multiplication definition is quite the same as the commutative property, they are not the same. Commutative property addresses whether the order of the terms will have any effect on the result. For example, operations such as function composition and matrix multiplication are associative but usually not commutative.

The easiest formula for understanding the associative property of multiplication is (a × b) × c = a × (b × c). This expression helps us realize that the placement of brackets is immaterial for the final result. Grouping of terms helps to make smaller components that make the multiplication process easier.

Associative binary operations deliver the same result regardless of how we place the pair of parentheses in the expression. For example, we can write a product of four elements without changing the order of factors in five possible ways:

As the product operation is associative, the law states all the above formulas yield the same result. Unless the formula without the parenthesis has a different meaning, it can be considered unnecessary, and the net product can be simply written as abcd.

If we keep increasing the number of elements in the operations, the number of ways to place the parentheses also increases rapidly. However, ultimately they remain unnecessary for disambiguation.

An example where this does not work is the logical biconditional . It is associative, thus A↔ (B↔C) is equivalent to (A↔B) ↔C, but A↔B↔C most commonly means (A↔B and B↔C), which is not equivalent.

Here are some important features of the associative property of multiplication:

Let us understand the property of multiplication with example.

Consider the expression given below.

2 * 4 * 6

We’ll solve this expression by two different grouping methods.

For the first process, we will group the numbers 2 and 4. In the second process, we will group the numbers 4 and 6.

As we can see, we get the same result from both processes.

Let us look at another example to verify this property.

Consider the expression given below.

10 * 5 * 7

We’ll solve this expression by two different grouping methods.

For the first process, we will group the numbers 10 and 5. In the second process, we will group the numbers 5 and 7.

As we can see, we get the same result from both processes.

This time we will solve an expression that contains four numbers.

Consider the expression given below.

2 * 3 * 7 * 11

We’ll solve this expression by three different grouping methods.

For the first process, we will group the numbers 2 and 3. In the second process, we will group the numbers 3 and 7. At the last one, we’ll group 7 and 11.

As we can see, we get the same result from both processes.

Rational numbers also follow the associative property of multiplication.

Suppose ab,cd andef are rational, then the associative property of multiplication can be written as:

ab* cd* ef= ab* cd* ef

Consider the expression given below.

12 *34 *56

We shall solve it by two processes as we have done before.

As we can see, we get the same result from both processes.

As we mentioned before, just like for multiplication, the associative property is also valid under the addition operation.

For example, consider the expression: 2 + 5 + 10

Let us make two groups of numbers 2 and 5, and then another of 5 and 10.

Hence we have proved that the associative property for addition is perfectly valid.

As we mentioned before, the associative property does not hold for the subtraction operation. Let us prove it with a problem.

For example, consider the expression: 3 – 2 – 1

Let us make two groups of numbers 3 and 2, and another one of 2 and 1.

Since, 0≠2,

Hence, we have proved that associative property is not valid for the subtraction operation.

As we mentioned before, the associative property does not hold for the division operation. Let us prove it with a problem.

For example, consider the expression: 100 ÷ 10 ÷ 5

Let us make two groups of numbers 100 and 10, and another of 10 and 5.

Since, 2≠50,

Hence, we have proved that associative property is not valid for the division operation.

As we mentioned before, the associative property does not hold for the vector cross product operation. Let us prove it with a problem.

For example, consider the expression: i×i×j

Let us make two groups of unit vectors i and i, and another one of i and j.

Since 0 ≠-j,

Hence, we have proved that associative property is not valid for the vector cross product operation.

Ans) If three or more terms are multiplied together, we obtain the same end answer irrespective of how the terms are grouped.

a x (b x c) = (a x b) x c

Multiplication is an operation that has various properties. One of them is the associative property. This property tells us that how we group factors does not alter the result of the multiplication, no matter how many factors there may be. We begin with an example:

3 x 2 x 5

The associative property of multiplication says that if we first multiply 3 x 2 and multiply the result by 5, it would be the same as if we first multiplied 2 x 5 and afterward multiplied by 3.

(3 x 2) x 5 = 3 x (2 x 5)

Shall we check?

3 x 2 = 6

6 x 5 = 30

2 x 5 = 10

10 x 3 = 30

Do you see?  We have obtained the same result by multiplying in two different ways. This is the associative property of multiplication!

Let’s do it with another example:

2 x 3 x 4 x 5

We will multiply in a variety of ways to demonstrate the associative property of multiplication:

2 x 3 x 4 x 5

2 x 3 = 6

6 x 4 = 24

24 x 5 = 120

3 x 5 x 2 x 4

3 x 5 = 15

15 x 2 = 30

30 x 4 = 120

5 x 2 x 4 x 3

5 x 2 = 10

10 x 4 = 40

40 x 3 = 120

4 x 5 x 3 x 2

4 x 5 = 20

20 x 3 = 60

60 x 2 = 120

The associative property of multiplication is easy, right?

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a x (b x c) = (a x b) x c

Multiplication is an operation that has various properties. One of them is the associative property. This property tells us that how we group factors does not alter the result of the multiplication, no matter how many factors there may be. We begin with an example:

3 x 2 x 5

The associative property of multiplication says that if we first multiply 3 x 2 and multiply the result by 5, it would be the same as if we first multiplied 2 x 5 and afterward multiplied by 3.

(3 x 2) x 5 = 3 x (2 x 5)

Shall we check?

3 x 2 = 6

6 x 5 = 30

2 x 5 = 10

10 x 3 = 30

Do you see?  We have obtained the same result by multiplying in two different ways. This is the associative property of multiplication!

Let’s do it with another example:

2 x 3 x 4 x 5

We will multiply in a variety of ways to demonstrate the associative property of multiplication:

2 x 3 x 4 x 5

2 x 3 = 6

6 x 4 = 24

24 x 5 = 120

3 x 5 x 2 x 4

3 x 5 = 15

15 x 2 = 30

30 x 4 = 120

5 x 2 x 4 x 3

5 x 2 = 10

10 x 4 = 40

40 x 3 = 120

4 x 5 x 3 x 2

4 x 5 = 20

20 x 3 = 60

60 x 2 = 120

The associative property of multiplication is easy, right?

If you liked this post, share it with your friends so that they, too, can learn what the associative property of multiplication is.

Try Smartick for free!

Learn More:

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 $9 = 6$.