## Introduction

There are three most widely used properties of numbers are commutative, associative, and distributive properties. Let us know what distributive property means? The distributive property, also known as the distributive law of multiplication is the most frequently used property in mathematics. The name ‘Distribute’ itself implies that to divide something. Distributive property states that we have to divide the given operations on the numbers so that the given Mathematical equation becomes easier to solve.

## What is Distributive Property?

The distributive property is a rule of multiplication that is used in Mathematical operations namely addition and subtraction. In other words, the property states that

x(y + z) = xy + xz

(x + y)z = xz + yz

The distributive property is widely used for evaluating, simplifying, or expanding expressions.

## Distributive Property Definition

The definition of distributive property simply states that “multiplication distributed over addition.”

For instance, take the equation: i(j + k)

If we apply the distributive property here, we have to multiply the variable ‘x’ with both y and z and then add as shown below:

(i x j) + (i x k) = ij + ik

## Distributive Property Formula

The distributive property formula is expressed as:

**x (y + z) = xy + xz**

## Distributive Property Examples

**Example 1: **Expand the equation 3(a + 5)

**Solution:**

3(a + 5) = 3 x a + 3 x 5 = 3a + 15

**Example 2: **Simplify (3p – p²)4p³

**Solution:**

Applying distributive property, we get

(3p – p²)4p³ = (3p x 4p³) – (p²x 4p³)

= 12p^{4 } – 4p^{5}

## Commutative Property

Commutative law states that the digits on which we operate can be moved or swapped from their positions without making any difference to the result. This holds true for Addition and Subtraction, but not on Multiplication or Division.

## Commutative Property of Addition

The commutative property of addition states that when we add two numbers, the result obtained will be the same, even the position of the numbers gets interchanged. For example, if ‘A’ and ‘B’ are any two real numbers, then according to the commutative property of addition, it is true that **A + B = B + A**

**Example:** 9 + 3 = 12 or 3 + 9 = 12

Hence Proved

## Commutative Property of Multiplication

The commutative property of multiplication states that when we multiply two numbers, the result obtained will be the same, even the position of the numbers gets interchanged. For example, if ‘A’ and ‘B’ are any two real numbers, then according to the commutative property of addition, it is true that **A ** ** B = B ** ** A**

**Example:** 9 3 = 27 or 3 9 = 27

Hence Proved

## Commutative Property of Subtraction And Division

Commutuaive property is not valid for subtraction and division because the result obtained will not be same even when the orders of numbers gets changed in subtraction of division.

**Example: **For Subtraction X Y Y X

7 – 4 = 3

4 – 7 = -3

Hence, the result obtained is not same. This shows commutative property is not valid for multiplication.

Also,

For Division: X Y Y X

6 3 = 2

3 6 = ½

Hence, the result obtained is not same. This shows commutative property is not valid for division.