Associative Law

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.

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

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.

Ans:

(-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

Ans:

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:

1. 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 = 1269

a = 14

Commutative Associative Distributive Examples with Solutions

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

4. Solve 3(4 + 5) using the distributive property.

Solution: Using the distributive property formula,

“a × (b + c) = a × b + a × c”

Multiply the term on the outside by both the terms that are inside the parenthesis, we get,

= 3 × 4 + 3 × 5

= 12 + 15= 27

∴ the value of 3(4 +5) = 27.

5. Solve 10(12 + 15) using the distributive property formula.

Solution: Using the distributive property formula,

“a × (b + c) = a × b + a × c”

Multiply the term on the outside by both the terms that are inside the parenthesis, we get,

= 10 × 12 + 10 × 15

= 120 + 150 = 270

∴ the value of 10(12 + 15) = 270

6. If 2 × (3 × 5) = 30, then find (2 × 3) × 5 using associative property.

Solution: The associative property for any said set of three numbers (A, B, and C) expression can be expressed such as here (A × B) × C = A × (B × C)

Assumed = 2 × (3 × 5) = 30

By the associative property formula, we can assess (2 × 3) × 5.

To prove: (2 × 3) × 5 = 30 or not first, solve the terms that are inside of the parentheses.

= 6 × 5 = 30

∴ 2 × (3 × 5) = (2 × 3) × 5 = 30.

7. If 3 × (6 × 4) = 72, then find (3 × 6) × 4 by associative property.

Solution: Since multiplication gratifies the formula of associative property, (3 × 6) × 4 = 3 × (6 × 4) = 72

Ways to remember and make Notes of the Important Properties of Numbers

The elementary properties of real numbers, together with the associative, commutative, and distributive properties, are very important when it comes to learning about addition, multiplication, and so on. They are also the stepping stones for the beginning stages of algebra. Once you comprehend each property, many difficult mathematical problems can be solved easily. One of the best ways to remember each property is to distinguish them by their names as follows:

• You can associate the associative property with its name. The associative property defines how you can come up with different sets of numbers together by addition or multiplication with the same result. Recollect that in adding and multiplying, numbers or variables can associate in different groups with each other for the same result.

•  You can connect the commutative property to the term commute. Rendering to the commutative property, while multiplying or adding numbers or variables the order is not necessary. The numbers or variables can "commute" or move from one place to another and the outcome will exactly be the same.

• Think of the distributive property as giving out or distributing a number all over a quantity when multiplying. For instance, if you have 2(x+y) you can dispense the 2 to give the equation as here 2x+2y.