In mathematics, the associative property is a property of some dyadic operations which is a calculation that combines two elements to produce another element. We will further study associative property in case of addition and multiplication. The associative property is not valid in case of division and subtraction.

**What is the Associative Property?**

It states that one can add or multiply regardless of how the numbers are grouped. By ‘grouped’ we mean ‘how you use parenthesis’. In simple words, if you are adding or multiplying it does not matter where you put the parenthesis.

\(x+\left ( y+z \right )= \left ( x+y \right )+z\)

\(\left ( x \times y \right )\times z= x\times \left ( y\times z \right )\)

Even though the parentheses were rearranged on each line, the values of the expressions were not altered. As the above holds true when performing addition and multiplication on any real numbers, it can be said that “addition and multiplication of real numbers are associative operations”. Apart from this there are also many important operations that are non-associative; some examples include subtraction, exponentiation, and the vector cross product.

Source: en.wikipedia.org

**Generalized Associative Law**

The law says “If a binary expression or operation is associative, repeated application of the operation produces the same result regardless of how valid pairs of parentheses are inserted in the expression”. For example, we can write a product of four elements, without changing the order of the factors, in five possible ways:

\(\left ( \left ( ab \right )c \right )d\)

\(\left ( ab \right )\left ( cd \right )\)

\(\left ( a\left ( bc \right ) \right )d\)

\(a\left ( \left ( bc \right )d \right )\)

\(a\left ( b\left ( cd \right ) \right )\)

**Associative Property of Multiplication**

It says that the way in which factors are grouped in a multiplication problem does not change the product. For example, the expression \(5 \times 4 \times 7\)

Let’s start by grouping 5 and 4 together. So we can express it;

\(\left ( 5 \times 4 \right ) \times 7 = 20 \times 7\) =140

Now let’s group 4 and the 7 together,

\(5\times \left ( 4\times 7 \right ) = 5 \times 28= 140\)

Regrouping does not change the answer.

**Associative Property of Addition**

It states that the sum of three or more numbers remains the same regardless of how the numbers are grouped.

Below is the example of how the sum does not change irrespective of how we group the addends.

35+20+12

Let’s start by grouping 35 and 20 together. So we can express it;

\(\left ( 35+ 20 \right )+ 12\)

55+12= 67

Now let’s group 20 and the 12 together,

\(35+\left ( 20+12 \right )\)

35+32= 67

**Application of Associative Property**

The associative property helps in terms of making concrete from a combination of three ingredients: cement, gravel, and water. So, we get to know that adding together cement, gravel, and water to make concrete is not an associative process.

The Associative Property helps to Speed Up Arithmetic. The idea is that instead of multiplying a list of numbers in the order they’re written from left-to-right, you can multiply them in any order you want. It helps to save a lot of time and multiply numbers faster easier.

**Solved Examples for Associative Property**

Q.1. Prove Associative property in case of Addition for the below example:

15+10+4 = 29.

Ans- Lets regroup the numbers and calculate the result. As per the associative property we can say that;

\(15+\left ( 10+4 \right )= \left ( 15+10 \right )+4\)

\(15+\left ( 14 \right )= \left ( 25 \right )+4\)

29=29

L.H.S=R.H.S

Q.2. Prove Associative property in case of Multiplication for the below example

\(10\times 2\times 9=180\)

Ans- Lets regroup the numbers and calculate the result. As per the associative property we can say that;

\(10\times \left ( 2\times 9 \right )= \left ( 10\times 2 \right )\times 9\)

\(10\times \left ( 18 \right )= \left ( 20 \right )\times 9\)

180=180

L.H.S=R.H.S

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