 # Distributive Property

Distributive property explains that the operation performed on numbers, available in brackets that can be distributed for each number outside the bracket. It is one of the most frequently used properties in Maths. The other two major properties are commutative and associative property.

The distributive property is easy to remember. There are a number of properties in Maths which will help us to simplify not only arithmetical calculations but also the algebraic expressions.

## Distributive Property Definition

The Distributive Property is an algebraic property that is used to multiply a single value and two or more values within a set of parenthesis. The distributive Property States that when a factor is multiplied by the sum/addition of two terms, it is essential to multiply each of the two numbers by the factor, and finally perform the addition operation. This property can be stated symbolically as:

A ( B+ C) = AB + AC

Where A, B and C are three different values.

Let’s consider a simple example: 2(4 + 3).

Since the binomial “4 + 3” is in the parenthesis, according to the order of operations, you have to calculate the value of 4 + 3 and then multiply it by 2, which gives the resultant value as 14. ### Distributive Property with Variables

Consider an example here: 6(2+4x)

The two values inside the parenthesis cannot be added since they are not like terms, therefore it cannot be simplified any further. We need a different method and this is where Distributive Property can be applied.

If you apply Distributive Property,

6× 2 + 6 × 4x

The parenthesis no longer exists and every term is multiplied by 6.

Now, you can simplify the multiplication for individual terms.

12 + 24x

The distributive property of multiplication lets you simplify expressions wherein you multiply a number by a sum or difference. According to this property, the product of a sum or difference of a number is equal to the sum or difference of the products. In algebra, we can have the distributive property for two arithmetic operations such as:

• Distributive Property of Multiplication
• Distributive Property of Division

Here, we will discuss the most frequently used distributive property of multiplication over addition in detail.

## Distributive Property of Multiplication

The distributive property of multiplication over addition is applied when you multiply a value by a sum. For example, you want to multiply 5 by the sum of 10 + 3.

As we have like terms, we usually first add the numbers and then multiply by 5.

5(10 + 3) = 5(13) = 65

But, according to the property, you can first multiply every addend by 5. This is known as distributing the 5 and then you can add the products.

The multiplication of 5(10) and 5(3) will be performed before you add.

5(10) + 5(3) = 50 + 15 = 65

You can note that the result is the same as before.

You probably use this method without actually knowing that you are using it.

The below equations describe both the methods. We have 10 and 3 on the left-hand side then multiplied by 5. This expansion is rewritten by applying the distributive property on the right-hand side where we distribute 5 then multiply by 5 and add the results. You will see that the result is similar in each case.

5(10 + 3) = 5(10) + 5(3)

5(13) = 50 + 15

65 = 65

The distributive properties of addition and subtraction can be utilized to rewrite expressions for different purposes. When you multiply a number by a sum, you may add and multiply. Also, you can first multiply each addend and then add the products. This applies to subtraction as well. In every case, you disturb the outer multiplier to every value in the parenthesis, so that multiplication occurs with every value before addition or subtraction.

## Distributive Property of Division

We can divide larger numbers using the distributive property by breaking those numbers into smaller factors.

Let us see an example here:

Q: Divide 84 ÷ 6.

We can write 84 as 60+24

Hence,

(60 + 24) ÷ 6

Now distributing division operation for each factor in the bracket we get;

(60 ÷ 6) + (24 ÷ 6)

= 10 + 4

= 14

## Problems and Solutions

Example 1:

Solve the given expression using the distributive property:

(i) 4(2x4+ 7x)

(ii) 2x(x2+ y)

(iii) 4(7xy+ 13yx)

Solution:

According to the distributive property,

A ( B + C) = AB + AC

(i) 4(2x4+ 7x)

Using distributive law we have,

= 4. 2x4+ 4. 7x

= 8x4+ 28x

(ii) 2x(x2+ y)

Using Distributive property,

= 2x .  x + 2x. y

= 2x3+ 2xy

(iii) 4(7xy+ 13yx)

Using distributive property, we have

= 4. 7xy + 4. 13yx

= 28 xy + 52 xy

= 80 xy