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**Associative Property Formula**

**Introduction to Associative Property Formula**

The associative property of multiplication states that the way in which the numbers are grouped in a multiplication problem does not affect or change the product of those numbers. In other words, the product of three or more numbers remains the same irrespective of the way they are grouped. Let us study more about the associative property of multiplication in this article.

**What is the Associative Property of Multiplication?**

According to the associative property of multiplication, if three or more numbers are multiplied, we get the same result irrespective of how the three numbers are grouped. Here, grouping means the way in which the brackets are placed in the given multiplication expression. Observe the following example to understand the concept of the associative property of multiplication. The expression on the left-hand side shows that 6 and 5 are grouped together, whereas, the expression on the right-hand side groups 5 and 7 together. However, when we finally multiply all the numbers, the resultant product is the same.

**Associative Property of Multiplication Formula **

The formula for the associative property of multiplication is (a × b) × c = a × (b × c). This formula tells us that no matter how the brackets are placed in a multiplication expression, the product of the numbers remains the same. The grouping of numbers with the help of brackets helps to create smaller components which makes the calculation of multiplication easier. Observe the following formula for the associative property of multiplication.

Let us understand the formula using numbers. For example, let us multiply 2 × 3 × 4 and see how the formula of associative property of multiplication is proved with the help of the following steps:

- Step 1: Let us group 2 and 3 together making it (2 × 3) × 4. If we find the product of this expression, we get 6 × 4, which is 24.
- Step 2: Now, let us group 3 and 4 together making it 2 × (3 × 4). If we multiply this expression, it comes to 2 × 12, which again gives the product as 24.
- Step 3: This means that no matter how we group the numbers in a multiplication expression, the product remains the same.

**Associative Property of Multiplication and Addition**

The associative property states that multiplication and addition of numbers can be done irrespective of how they are grouped.

For example, to add 7, 6, and 3, if we group them as 7 + (6 + 3), the sum that we get is 16. Now, let us group it as (7 + 6) + 3 and we see that the sum is 16 again.

This is the associative property of addition which applies to multiplication as well.

For example,

let us multiply 7, 6, and 3 and group the numbers as 7 × (6 × 3).

The product of these numbers is 126.

Now, if we group the numbers as (7 × 6) × 3, we get the same product, that is, 126.

Observe the following figure which shows the associative property of multiplication and addition.

**Tips on the Associative Property of Multiplication:**

Here are a few important points related to the associative property of multiplication:

- The associative property always applies to 3 or more numbers.
- The associative property exists in addition and multiplication and cannot be applied to subtraction and division.

**Solved Examples on Associative Property of Multiplication**

**Example 1:** Which of the two expressions are equivalent to 8 × 3 × 4?

a.) (8 × 3) × 4

b.) 24 × 4

c.) 11 × 4

Solution:

The product of the given expression is 8 × 3 × 4 = 96. Now, let us check the product of the following expressions.

a.) The product of (8 × 3) × 4 is 96.

b.) The product of 24 × 4 is 96.

c.) The product of 11 × 4 is 44.

Therefore, the first two expressions are equivalent to 8 × 3 × 4. For the first expression, we used the associative property of multiplication to group together 8 and 3, and the second option is the simplified form of the first option. So, both are correct.

**Example 2:** Choose the correct number to fill the blank in the expression: 5 × (4 × 3) = (5 ×___) × 3

a.) 3

b.) 4

c.) 5

Solution:

The associative property of multiplication states that a × (b × c) = (a × b) × c.

So, after substituting the given equation in this formula, we get 4 as the answer.

The correct option is (b) 4 which means that the product of both the sides will be equal to 60 if we place 4 in the blank.

**Example 3:** Fill the missing number in the blank.

10 × (8 × 7) = (10 × 8) × ___

Solution:

According to the associative property of multiplication: a × (b × c) = (a × b) × c. Substituting the values in the formula: 10 × (8 × 7) = (10 × 8) × 7

Hence, the missing number will be 7 because the product of both the expressions is equal to 560.

**Frequently Asked Questions on Associative Property of Multiplication**

1: What is the Associative Property of Multiplication in Math?

Answer: The associative property of multiplication states that the product of three or more numbers remains the same regardless of how the numbers are grouped. For example, 3 × (5 × 6) = (3 × 5) × 6. Here, no matter how the numbers are grouped, the product of both the expressions remains 90.

2: What is the Associative Property of Multiplication Formula?

Answer: The formula for the associative property of multiplication is written as

a × (b × c) = (a × b) × c. This means that the grouping of any three or more numbers does not affect their product.

3: What is the Associative Property of Multiplication and Addition?

Answer: The associative property applies to addition and multiplication which means that the addition and multiplication of numbers can be done irrespective of the way they are grouped. The associative property of addition is written as: a + (b + c) = (a + b) + c, which means that the sum of any three or more numbers does not change even if the grouping of the numbers is changed. Similarly, the associative property of multiplication is written as: a × (b × c) = (a × b) × c, which means that the product of any three or more numbers remains the same even after they have been grouped in a different way.

4: Give an Example of the Associative Property of Multiplication.

Answer: The associative property of multiplication can be understood with the help of an example of any three numbers. If we multiply (4 × 2) × 10, we get the product as 8 × 10 = 80. Now, if we group these numbers as 4 × (2 × 10), we still get the product as 4 × 20 = 80. This proves the associative property of multiplication.

5: What is the Associative Property of Multiplication of Whole Numbers?

Answer: The associative property of multiplication of whole numbers says that the product of three or more whole numbers does not change even if the numbers are grouped in a different way.

For example, 11 × (5 × 2) = (11 × 5) × 2. Here, the product of both the expressions is 110.

6: What is the Difference Between the Commutative and the Associative Property of Multiplication?

Answer: The commutative property of multiplication states that changing the order of numbers does not change the product of the given numbers. For example, 6 × 8 = 8 × 6 = 48. The associative property of multiplication states that changing the grouping of numbers does not change the product of the given numbers. For example, 7 ×(2 × 3) = (7 × 2) × 3 = 42.

7: What is the distributive property of rational numbers?

Answer: The distributive property states, if a, b and c are three rational numbers, then;

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

8: The commutative property of rational number is applicable to addition and multiplication only. True or false?

Answer: True. The commutative property of rational numbers is applicable for addition and multiplication only and not for subtraction and division.