# Associative and Commutative Property,Distributive Property

For rewriting and manipulating some algebraic expressions. It’s key to know about the associative and commutative property, along with the distributive law.

The ‘Associative Property’ is something that applies to the operations of addition and multiplication.

Being that you can add or multiply values in any order, and it doesn’t matter how the numbers are grouped together.

## Associative and Commutative Property

### Associative Property:

( a + b ) + c   =   a + ( b + c )

( 2x + 4x ) + 3x \space = \space 6x + 3x \space = \space 9x

2x + ( 4x + 3x ) \space = \space 2x + 7x \space = \space 9x

Associative property with multiplication is:
( a × b ) × c   =   a × ( b × c )

( 2x \times 4x ) \times 3x \space = \space 8x \times 3x \space = \space 24x^2

2x \times ( 4x \times 3x ) \space = \space 2x \times 12x \space = \space 24x^2

The property does NOT apply to division or subtraction however.

( 8x \div 4x ) \div 2x \space = \space 2x \div 2x \space = \space 1

8x \div ( 4x \div 2x ) \space = \space 8x \div 2 \space = \space 4x

( 8x \space – \space 4x ) \space – \space 2x \space = \space 4x \space – \space 2x \space = \space 2x

8x \space – \space ( 4x \space – \space 2x ) \space = \space 8x \space – \space 2x \space = \space 6x

### Commutative Property:

a + b   =   b + a

5x + 3x \space = 8x

3x + 5x \space = \space 8x

Commutative property with multiplication is:
a × b   =   b × a

3x \times 4x \space = \space 12x

4x \times 3x \space = \space 12x

The commutative property also does NOT apply to division or subtraction however.

5x \space – \space 3x \space = 2x

3x \space – \space 5x \space = \space {\text{-}}2x

8x \div 4x\space = \space 2

4x \div 8x \space = \space {\frac{1}{2}}

## Distributive Property

Following on from the associative and commutative property is the distributive property.

The distributive property with multiplication over addition is:
a × ( b + c )  =  ( a × b ) + ( a × c )

2x \times ( 3x + x ) \space = \space 2x \times 4x \space = \space 8x^2

( 2x \times 3x ) + ( 2x \times x ) \space = \space 6x^2 + 2x^2 \space = \space 8x^2

The distributive property with multiplication over subtraction is:
a × ( bc )   =   ( a × b ) − ( b × c )

2x \times ( 3x – x ) \space = \space 2x \times 2x \space = \space 4x^2

( 2x \times 3x ) \space – \space ( 2x \times x ) \space = \space 6x^2 – 2x^2 \space = \space 4x^2

The distributive property though doesn’t work with division over division.

10x \div ( 4x \div 2x ) \space = \space 10x \div 2x \space = \space 5

( 10x \div 4x ) \div ( 10x \div 2x ) \space = \space {\frac{5}{2}} \div 2 \space = \space {\frac{5}{4}}

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