Associative Law

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Associative Commutative and Distributive Properties

There are four different properties of numbers namely, associative, commutative, multiplicative, and identity.  You should be familiar with these properties as there are many times in algebra when you are asked to simplify an expression.  These properties will help you to solve complicated algebra problems easily. The associative, commutative, and distributive properties are commonly used to simplify the algebraic expression.

Here, we will discuss the commutative, associative, and distributive properties in detail.


Associate Law Definition

The associative law definition states that when any three real numbers are added or multiplied, then the grouping (or association) of the numbers does not affect the result. For example, when we add: (a + b) + c = a + (b + c), or when we multiply : (a x b) x c = a x (b x c).

While associative laws hold for ordinary mathematics with real numbers or imaginary numbers, there are certain applications such as nonassociative algebras- in which the law does not hold.


Associative Law of Addition

Below are the two ways of simplifying and solving additional problems.

3 + 4 + 5 = 7  + 5 = 12

Here, a similar problem is solved but 4 is added to 5 to make 9. Solving addition in this way will also yield the same answer.

3 + 4 + 5 = 3 + 9 = 12

The associate law of addition states that the numbers, will adding, can be regrouped using parentheses.  In the following expression, the parentheses are used to group numbers so that you know what to add first.  Note that when parentheses are given, any numbers within the parentheses are numbers that will be added first. The expression can we write using the associative laws as follows:

(3 + 4) + 5 = 7 + 5 = 12

3 + (4 + 5) = 3 + 9 = 12

Here, it is clear that the parentheses do not affect the final answer. The final answer will be the same regardless of where the parenthesis is.


Associative Law of Multiplication

The associative law of multiplication is the same as the associative law of addition. It states that no matter how you group the numbers you are multiplying together, the answer will always be the same. The associative property of multiplication says:

(xy)z = x(yz)


Example:

(5 x 7) x 3 = 35 x 3 = 105

5 x (7 x 3) = 5 x 21 = 105


Associative Law of Vector Addition

The associative law of vector addition states that the sum of the vectors remains the same regardless of the order or grouping in which they are arranged.

Consider the following three vectors:

\[\vec{A}\], \[\vec{B}\], and \[\vec{C}\] 

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Appy head to tail rule to get the resultant of (\[\vec{A}\] + \[\vec{B}\]) and (\[\vec{B}\] + \[\vec{C}\])

At last, find the resultant of these three vectors again as shown below:

\[\bar{OR}\] = \[\bar{OP}\] + \[\bar{PR}\]

Or 

\[\vec{R}\] = \[\vec{A}\] + (\[\vec{B}\] + \[\vec{C}\])

And

\[\bar{OR}\] = \[\bar{OQ}\] + \[\bar{QR}\]

\[\vec{R}\] + (\[\vec{A}\] + \[\vec{B}\]) + Hence, from equation (1) and (2), we get

\[\vec{A}\] + (\[\vec{B}\]) + \[\vec{C}\]) = (\[\vec{A}\] + \[\vec{B}\]) + \[\vec{C}\]


Commutative Properties 

The commutative property states that the numbers on which we operate can be moved or swapped in any position without making any difference to the answers. The commutative property holds for both addition, and multiplication, but not for subtraction and division.


Commutative Property of Addition

If a and b are real numbers, then

a + b = b + a


Commutative Property of Multiplication

If a and b are real numbers, then

ab = ba

This commutative property of multiplication also works for more than 2 numbers i.e.

A x b x c x d = d x c x b x a


Commutative and Associative Property of Addition Example

1. Write the Expression (-14.5) + 24.5 in a Different Way Using the Commutative Property of Addition and Show that the Result of Both the Expressions Has the Same Answer.

Solution:

(-14.5) + 24.5 = 10            (Adding)

(24.5) + (- 14.5) = 10   (Using the commutative property, you can swap -14.5 and 24.5 so that they are in different order).

(24.5) + (- 14.5) = 10  (Adding 24.5 and -14.5 is the same as subtracting 14.5 from 24.5. The sum is 10. 

24.5 - 14.5 = 10

Answer: (-14.5) + 24.5 = 10 and (24.5) + (- 14.5) = 10


2. Show That the Following Numbers Follow the Associative Property of Addition:

3, 6, and 8

Solution:

3 + 6 + 8

( 3 + 6 ) + 8 = 9 + 8 = 17

Or

3 + ( 6 + 8) = 3 + 14 = 17

The result is the same in both cases. Hence,

( 3 + 6 ) + 8 = 3 + ( 6 + 8)


Distributive Property

The distributive property is a rule that relates to the addition and multiplication

a(b + c) = ab + ac

(a + b)c = ac + bc

It is a useful property for expanding expressions, evaluating expressions, and simplifying expressions.

Let us understand with an example:

Example:

Solve the following equation using the distributive property:

9(a - 5) = 81

Solution:

Step 1: Find the product of a number with the numbers given in parenthesis as shown below:

9(a) - 9(5) = 81

9a - 45 = 81

Step 2:  Arrange the numbers in such a way that constant terms and the variable terms are on the opposite of the equation.

9a - 45 - 45 = 81 + 45

9a  = 126

Step 3: Solve the equation

9a = 126

a = \[\frac{126}{9}\]

a = 14


Commutative Associative Distributive Examples with Solution

Here are some of the commutative associative distribution examples with solutions to make you understand the concept better.


1. Solve the Following Using the Distributive Property.

(7a + 4)²

Step 1: Expand the equation 

(7a + 4)² = ( 7a + 4) ( 7a + 4)

Step 2: Find the product

( 7a + 4) ( 7a + 4) = 49a² + 28a + 28a + 16

Step 3: Add all the like terms together

49a² + 56a + 16


2. Show That the Following Numbers Follow the Commutative Property of Multiplication

3, 4, 6, and 8

Solution:

As we know, If a b, c, and c are real numbers, then

a x b x c x d =  d x c x b x a

Accordingly:

L.H.S. = 3 x 4 x 6 x 8 = 576

R.H.S = 8 x 6 x 4 x 3 = 576

The result is the same in both the case 

3 x 4 x  6 x 8 = 8 x 6 x 4 x 3 = 576


3. Hitesh Knows That 6 x 2 = 12. His Teacher Asked Him to Find the Value of 6 x 2 x 3 Using the Associative Property of Multiplication. Can You Help Hitesh to Find the Right Answer?

Solution:

As we know the associative property of multiplication says that 

6 x 2 x 3  = (6 x 2) x 3

From the information available to Hitesh, we can say that

( 6 x 2) x 3 =  12 x 3 

Hence, the right answer is

12 x 3 = 36

∴ The answer is 36

FAQ (Frequently Asked Questions)

Q1. How Do the Commutative and Associative Properties Differ?

Ans. The commutative and associative properties are quite similar and can be easily mixed up. For this reason, it is important to understand the difference between the two. The commutative property discusses the order of certain mathematical operations. For a binary operation, one that includes only two elements- can be represented by the equation as a + b = b + a. The operation is commutative because the order of elements does not affect the result of the operation.


On the other hand, the distributive property discusses the grouping of elements in an operation. This can be represented by the equation (a + b) + c  = a + (b + c). The grouping of elements as represented by parentheses does not affect the results.

Q2. What Is the Distributive Property of Multiplication States?

Ans. According to the distributive property of multiplication, when any number is multiplied by the sum of any two numbers, the first number can be distributed to both of these numbers and multiplied by each of them separately, then adding the two products together for the same results as multiplying the first number by the sum. For example : 2 x (3 + 5) = 2 x 3 + 2 x 5.

Q3. How Do Commutative Associative Distributive Properties Help?

Ans. The commutative associative distributive properties help to rewrite the complicated expressions that are easier to deal with. When the expression is being rewritten using the commutative property, the order of numbers being added or multiplied gets changed whereas When an expression is rewritten using the associative property, the different pairs of numbers are grouped using the parenthesis.  


The commutative and associative property can be used to regroup and reorder any number in the expression as long as the expression is entirely made up of addends or factors, but not a combination of them.  On the other hand, the distributive property states that when you multiply a number by a sum, you can add and then multiply. You can also multiply each addend first and then add the product together. The same rules also apply if you multiply a number by its difference. 

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