Associative Property Formula

The associative property of mathematics is a rule that elaborates the importance of the grouping of numbers during mathematical operations. The associative property of mathematical operations is the property of numbers that says that the total result of the operations will not change however the numbers are arranged while adding, subtracting or multiplying. Here arranged implies that 'the way we used brackets'. In other words, if we add three or more numbers we come to the same answer irrespective of how the numbers are arranged or parenthesized in any order. 

The associative property is one of the three major properties of mathematical operations and particularly for addition. In this article, we will learn the associative property formula, the formula of associative property and the associative property of multiplication formulas with solved problems.

Formula of Associative Property

Mathematics is all about playing with numbers with numerous mathematical operations such as addition, multiplication, subtraction etc… As with every other operation, even these fundamental operations are governed by a set of rules. The associative property is one such important rule that had to be followed while addition and multiplication. We can perform multiplication and addition of the numbers in an equation irrespective of the arrangement or how the brackets have been used between the numbers and the formula used for this rule is known as the associative law formula. 

So, the associative property is only valid for the addition and multiplication operations. Corresponding associative property formulae are known as the associative property of multiplication formula and the associative property of addition formula. Let us learn the formula for associative property with a few solved examples.

Associative Property of Multiplication Formula

The associative property of multiplication is the property of numbers that states the multiplication of three or more numbers will not alter however the numbers are assorted while adding. In other words, if we multiply three or more numbers we will finally arrive at the exact answer irrespective of how the numbers are assorted or how the brackets are used. The associative law is one of the major properties of multiplication.

Let us now have a look at the associative property of multiplication formula. Consider three numbers a, b, and c. Now, the associative property states that the multiplicative product of three or more numbers remains the same irrespective of the way numbers are arranged or grouped. Mathematically, we write:

⇒(a x b) x c = a x (b x c)…….(1)

Equation (1) is known as the associative property of multiplication formula.

Associative Property of Addition Formula

The associative property of Addition is the property of numbers that states the Addition of three or more numbers will not alter however the numbers are assorted while adding. In other words, if we add three or more numbers we will finally arrive at the exact answer irrespective of how the numbers are assorted or how the brackets are used. The associative law is one of the major properties of Addition.

Let us now have a look at the associative property of addition formula. Consider three numbers a, b, and c. Now, the associative property states that the sum of three or more numbers remains the same irrespective of the way numbers are arranged or grouped. Mathematically, we write:

⇒(a + b) + c = a + (b + c)…….(2) 

Equation (2) is known as the associative property of addition formula.

Note

Important points to be remembered while using the associative property:

• The associative formula is valid only for addition and multiplication.

• Associative properties are in good agreement with the ability to associate or group numbers, which is not can not be done in the case of subtraction and division.

• The associative law is among the list of properties in mathematics that are helpful in the manipulation of mathematical equations to solve and arrive at the desired solutions.

Examples

Let us learn the associative property formula with some examples for better understanding.

1. Show that: 9 + (7 + 8) = (9 + 7) + 8.

Sol:

To prove the given expression, we must obtain the answer for the right-hand side and the left-hand side. Let us solve this question step by step.

Step 1: 

Write down the left-hand side of the equation i.e., 9 + (7 + 8) =?

Step 2: 

Add the numbers in the given bracket to obtain one number 9 + 15.

Step 3: 

Add these numbers and get a single number to the left-out numbers 24.

Step 4: 

Now, repeat the same process with the right-hand side of the equationi.e.,(9 + 7) + 8 =?

Step 5: 

Add the numbers in the bracket i.e., 16 + 8.

Step 6: 

Add this single number to the remaining numbers and arrive at the answer for it i.e., 24.

Therefore, from step 3 and step 6 it is proved that the LHS is equal to the RHS and therefore associative property is proved.

2. Show that 5 x (7 x 8) = (5 x 7) x 8.

Sol:

To prove the given expression, we must obtain the answer for the right-hand side and the left-hand side. Let us solve this question step by step.

Step 1: 

Write down the left-hand side of the equation i.e., 5 x (7 x 8) =?

Step 2: 

Find the product of the numbers in the given bracket to obtain one numberi.e., 5 x 56.

Step 3: 

Multiply these numbers and get a single number to the left-out numbers i.e., 280.

Step 4: 

Now, repeat the same process with the right-hand side of the equationi.e.,(5 x 7) x 8=?

Step 5: 

Find the product of the numbers in the bracket i.e., 35 x 8.

Step 6: 

Add this single number to the remaining numbers and arrive at the answer for it i.e., 280.

Therefore, from step 3 and step 6 it is proved that the LHS is equal to the RHS and therefore associative property is proved.

3. Show That Associative Property is Not Valid For Subtraction With a Suitable Example.

Sol:

The associative property is only valid for addition and multiplication. It can not be used for subtraction. Let us understand this with an example.

For ex.:

5 - (7 + 9)(5 - 7) + 9

To prove the given expression, we must obtain the answer for the right-hand side and the left-hand side. Let us solve this question step by step.

Step 1: 

Write down the left-hand side of the equation i.e., 5 -(7 + 9) =?

Step 2: 

Add the numbers in the given bracket to obtain one number i.e., 5 - 16

Step 3: 

Add these numbers and get a single number to the left-out numbers -11.

Step 4: 

Now, repeat the same process with the right-hand side of the equationi.e.,(5 - 7) + 9 =?

Step 5: 

Add the numbers in the bracket i.e., -2 + 9.

Step 6: 

Add this single number to the remaining numbers and arrive at the answer for it i.e., 7.

Therefore, from step 3 and step 6 it is proved that the LHS is NOT EQUAL to the RHS and therefore associative property is not valid for subtraction.