Associative Property of Multiplication

Associative Property is a fundamental concept in mathematics, particularly in multiplication. It is one of the three basic properties of multiplication, alongside the commutative and distributive properties. Understanding this property is crucial for mastering multiplication and for further mathematical studies.

What is Associative Property of Multiplication?

Associative Property of multiplication is a fundamental concept in mathematics. It states that the grouping of numbers in a multiplication operation does not affect the product of the numbers.

Associative Property of Multiplication

Associative Property of Multiplication Formula

Associative Property of Multiplication can be formally expressed as:

(a × b) × c = a × (b × c)

where a, b, c are any real numbers

Let’s consider an example to illustrate this property. Suppose we have the numbers 2, 3, and 4. According to the associative property:

(2 × 3) × 4 = 2 × (3 × 4)

If we calculate the left-hand side of the equation, we first multiply 2 and 3 to get 6 and then multiply 6 by 4 to get 24. On the right-hand side, we first multiply 3 and 4 to get 12 and then multiply 2 by 12 to also get 24. As you can see, the product is the same regardless of how the numbers are grouped.

Associative Property of Multiplication and Addition

Associative property is explained for both addition and multiplication operations and both of them are explained below,

Associative Property of Multiplication

The associative property asserts that the grouping of numbers in a multiplication expression does not affect the result. It is expressed as

(a × b) × c = a × (b × c)

It allows the rearrangement of factors without altering the product.

Associative Property of Addition

Similar to multiplication, the associative property for addition states that the grouping of numbers in an addition expression doesn’t impact the sum. It is represented as

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

It allows the rearrangement of addends without changing the result.

Learn more about, Associative Property

Associative Property of Subtraction

Associative Property is not true for Subtraction i.e

(A – B) – C ≠ A – (B – C)

Associative Property of Division

Associative Property is not true for Subtraction i.e.

(A ÷ B) ÷ C ≠ A ÷ (B ÷ C)

Associative Property of Matrix Multiplication

Associative Property is also followed in case of Matrix Multiplication. For three matrices A, B and C, associative property of matrix multiplication is given as

(A × B) × C = A × (B × C)

Learn, Matrix Multiplication

Associtive Property of Multiplication Conclusion

In conclusion, Associative Property of Multiplication is a fundamental mathematical principle that states the product of a set of numbers remains the same regardless of how they are grouped. Understanding this property is essential for mastering multiplication and for further mathematical studies.

Associative Property of Multiplication Example

Some example on Associative Property of Multiplication are,

Example 1: Using the associative property, solve the multiplication of three numbers: (2 × 3) × 4.

Solution:

(2 × 3) × 4 = 6 × 4 = 24

Now, let’s rearrange the grouping:

2 × (3 × 4) = 2 × 12 = 24

The result is the same in both cases, illustrating the associative property.

Example 2: Apply the associative property to the multiplication of 1, 5, and 2

Solution:

(1 × 5) × 2 = 5 × 2 = 10

And when the grouping is changed:

1 × (5 × 2) = 1 × 10 = 10

Once again, the result is consistent, affirming the associative property.

Example 3: Apply the Associative property to a set of decimals: 0.5 × 2 × 3

Solution:

(0.5 × 2) × 3

(0.5 × 2) × 3 = 1 × 3 = 3

Change the grouping

0.5 × (2 × 3) = 0.5 × 6 = 3

The result is unchanged, emphasizing the associative property.

Example 4: Demonstrate the associative property with : (-2 × 4) × 3.

Solution:

(-2 × 4) × 3 = -8 × 3 = -24

Rearrange the grouping

-2 × (4 × 3) = -2 × 12 = -24

The consistency holds, even with negative numbers.

Practice Questions on Associative Property of Multiplication

Some practice questions on Associative Property of Multiplication are,

Q1: Evaluate the expression using the associative property: (2 × 3) × 4

Q2: Simplify the following expression by applying the associative property: 5 × (7 × 2)

Q3: Show the steps in rearranging the numbers using the associative property: 4 × (6 × 3)

Q4: Apply the associative property to find the result: (9 × 2) × 5

Q5: Given the expression 3 × (2 × 4), demonstrate the use of the associative property to compute the result.

Associative Property of Multiplication – FAQs

What is Associative Property of Multiplication?

Associative property of multiplication states that the product of a set of numbers remains the same regardless of how they are grouped. Formally, for any real numbers a, b, and c, it can be expressed as: (a × b) × c = a × (b × c)

Why is Associative Property Important?

Associative property is important because it allows us to rearrange terms in a multiplication problem without changing the product. This flexibility can simplify complex calculations and make mental arithmetic easier.

Does Associative Property apply to Subtraction or Division?

No, the associative property does not apply to subtraction or division. For example, (a – b) – c ) is not always equal to a – (b – c), and (a / b) / c is not always equal to a / (b / c).

How is Associative Property Different from Commutative Property?

• Associative property refers to the grouping of numbers, while the commutative property refers to the order of numbers.
• Commutative property states that the order of numbers does not change the result in addition or multiplication.

What is an Example of Associative Property?

Take numbers 2, 3, and 4. According to the associative property:

(2 × 3) × 4 = 2 × (3 × 4)

Both sides of the equation give the same result, 24, thus, associative property is verified.