Algebraic Properties

The structural laws — commutativity, associativity, distributivity — that govern how operations behave.

Three algebraic laws let you move, group, and distribute terms without changing value.

Commutative

order can move

a+b=b+a

Associative

grouping can move

(a+b)+ c = a +(b+c)

Distributive

multiplication spreads

a(b + c)=ab + ac

Definition

There are three core properties that let you rearrange and regroup numbers and expressions freely:

Commutative — order doesn't matter: $a + b = b + a \qquad a \times b = b \times a$

Associative — grouping doesn't matter: $(a + b) + c = a + (b + c) \qquad (a \times b) \times c = a \times (b \times c)$

Distributive — multiplication distributes over addition: $a(b + c) = ab + ac$

Using commutativity to simplify mental arithmetic

$37 + 58 + 3 = 37 + 3 + 58 = 40 + 58 = 98$

Reordering to make a friendly pair.

Using distributivity to expand

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

Subtraction and division are NOT commutative

$8 - 3 \neq 3 - 8$ and $8 \div 4 \neq 4 \div 8$ . These properties apply to addition and multiplication only.

Try it

Use the distributive property to expand $4(2x - 5)$ .

Solution

$4(2x - 5) = 8x - 20$

Related concepts

Needs first