Properties of Operations
Apply commutative, associative, distributive, identity, and inverse properties to simplify expressions. Use the distributive property to expand and factor simple expressions.
A lesson of Pre-Algebra, in Expressions & Equations, about 10 minutes.
By the end of this lesson, you will be able to identify and apply the commutative, associative, distributive, identity, and inverse properties.
The Properties
| Property | Addition | Multiplication |
|---|---|---|
| Commutative | $a + b = b + a$ | $ab = ba$ |
| Associative | $(a+b)+c = a+(b+c)$ | $(ab)c = a(bc)$ |
| Identity | $a + 0 = a$ | $a \times 1 = a$ |
| Inverse | $a + (-a) = 0$ | $a \times \frac{1}{a} = 1$ ($a \neq 0$) |
| Distributive | $a(b+c) = ab + ac$ | |
| Zero | — | $a \times 0 = 0$ |
Using Properties to Simplify
Commutative: rearrange terms to make computation easier.
$37 + 48 + 63 = 37 + 63 + 48 = 100 + 48 = 148$ ← faster!
Associative: regroup to simplify.
$25 \times 17 \times 4 = 25 \times 4 \times 17 = 100 \times 17 = 1700$
Distributive: factor out a common value.
$6(14) = 6(10 + 4) = 60 + 24 = 84$
$3x + 3y = 3(x + y)$ (factoring — reverse distributive)
Identifying Properties
"$3 \times (4 \times 7) = (3 \times 4) \times 7$" → Associative (multiplication)
"$5(x + 2) = 5x + 10$" → Distributive
"$8 + 0 = 8$" → Identity (addition)
Summary
Commutative: order doesn't matter for + and ×.
Associative: grouping doesn't matter for + and ×.
Distributive: $a(b+c) = ab + ac$; the key algebraic property.
Identity: $+0$ and $\times 1$ leave values unchanged.
Inverse: opposites/reciprocals give identity elements (0 or 1).