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# Associative Property

The associative property states that the addition and multiplication of numbers are possible regardless of how they are grouped. By grouping, we mean the numbers that are provided in parentheses.

For example, say you were going to add the numbers 3, 7, and 12. The answer would be the same whether you wrote the equation as $\left(3+7\right)+12$ or $3+\left(7+12\right)$ . The same is true of multiplication. If you were to multiply $4×2×5$ , the answer would be the same whether it was written as $4*\left(2*5\right)$ or $\left(4*2\right)*5$ .

## Properties of mathematical equations

Mathematical equations each have their own manipulative properties. These properties, or principles, are helpful in solving such equations. There are three basic properties that comprise the backbone of mathematics. These three properties are used to perform a variety of arithmetic operations. They are the:

## Associative property definition

As the name implies, the associative property involves grouping. The origin of the term comes from the word "associate." The basic mathematical operations that are subject to the associative property are addition and multiplication. The property is applied when there are more than two numbers. The property does not pertain to subtraction or division problems.

As is the case of the commutative property, the order of the grouping doesn't matter. The result will not be changed as the order of the numbers is changed. Therefore, the bottom line of the associative property is that it doesn't matter which part of the operation is carried out first-the answer will be the same no matter what.

Note that both the associative and commutative properties are applicable to addition and multiplication only.

## The associative property of addition

Addition follows the associative property such that, regardless of how numbers are placed in parentheses, the final sum of the numbers will be the same. The associative property of addition states that:

$\left(p+q\right)+r=p+\left(q+r\right)$

Example 1

Say we want to add $6+8+10$ . We can see that the answer is 24. Let's see what we get by grouping numbers in parentheses.

$\left(6+8\right)+10$

First, we perform the addition in parentheses.

$=14+10$

$=24$

Now let's see if we group them in a different way.

$6+\left(8+10\right)$

Again, we perform the addition in the parentheses first, and we get a different equation.

$=6+18$

$=24$

With the numbers grouped, we come to the final answer a different way, but we come to the same answer either way.

We can see the property in action in equalities as well. Let's try one now.

Example 2

$3+\left(2+1\right)=\left(3+2\right)+1$

Perform the addition in the parentheses on each side, then complete the solution using addition.

$3+3=5+1$

$6=6$

## Associative property of multiplication

The associative property of multiplication states that:

$\left(xy\right)z=x\left(yz\right)$

Example 3

Let's solve the following equation two different ways.

$3*4*5$

Start by grouping the first two numbers.

$\left(3*4\right)*5$

First, perform the multiplication in parentheses.

$=12*5$

$=60$

Now let's solve it another way.

$3*\left(4*5\right)$

Again, we'll perform the multiplication in parentheses first.

$=3*20$

$=60$

We get the same answer either way because of the associative property of multiplication. Let's try one more example.

Example 4

Solve the following in two different ways.

$2*\left(-6\right)*4$

First, we'll group the first two numbers.

$\left(2*\left(-6\right)\right)*4$

$=-12×4$

$=-48$

Now let's solve the problem with the numbers grouped another way.

$2*\left(-6*4\right)$

$=2*\left(-24\right)$

$=-48$