Properties of Operations and Equivalent Expressions

The rules that say what is allowed

Rearranging an expression is only safe if a property permits it. There are four worth naming.

Property	Addition	Multiplication
commutative	$a + b = b + a$	$ab = ba$
associative	$(a + b) + c = a + (b + c)$	$(ab)c = a(bc)$
identity	$a + 0 = a$	$a \times 1 = a$
distributive	—	$a(b + c) = ab + ac$

Commutative: order does not matter

$$3 + 8 = 8 + 3 \qquad 4x = x \cdot 4$$

Both give the same answer, so terms can be written in any order.

It applies to addition and multiplication only. Subtraction and division are not commutative:

$$7 - 3 = 4 \qquad \text{but} \qquad 3 - 7 = -4$$ $$10 \div 2 = 5 \qquad \text{but} \qquad 2 \div 10 = 0.2$$

There is a way around that, and it matters later. Rewrite the subtraction as adding the opposite. It is now an addition, so the order can be swapped:

$$7 - 3 = 7 + (-3) = (-3) + 7$$

That rewrite is what lets terms move freely when collecting like terms.

Associative: grouping does not matter

$$(2 + 5) + 8 = 2 + (5 + 8)$$

Both give $15$. The brackets move; the order stays.

$$(3 \times 4) \times 5 = 3 \times (4 \times 5) = 60$$

This is what lets you add a long list in any order you like. Adding $17 + 8 + 2$ is quicker as $17 + (8 + 2) = 17 + 10$. The property is what makes that regrouping a rule rather than a lucky shortcut.

Distributive: multiplying across a bracket

$$a(b + c) = ab + ac$$ $$3(x + 4) = 3x + 12$$

This is the only property linking multiplication with addition. It gets a lesson of its own, because so much rests on it.

Equivalent expressions

Two expressions are equivalent when they give the same value for every input.

$$2(x + 3) \quad \text{and} \quad 2x + 6$$

$x$	$2(x + 3)$	$2x + 6$
$0$	$6$	$6$
$1$	$8$	$8$
$5$	$16$	$16$
$-4$	$-2$	$-2$

They agree everywhere. The distributive property explains why: one is the other rearranged by a rule that keeps the value the same.

Why testing values cannot prove equivalence

This is the point worth being careful about.

Take $x + x$ and $x^2$. At $x = 2$:

$$2 + 2 = 4 \qquad 2^2 = 4$$

They match. They are not equivalent, and $x = 3$ shows it:

$$3 + 3 = 6 \qquad 3^2 = 9$$

One agreement was a coincidence. Equivalence needs agreement at every value. No amount of testing covers every value, so testing can never prove it.

What testing does do is disprove equivalence, and one case is enough. That makes it a fast error-check. After simplifying, put a value into the original and into your answer. If they differ, something went wrong. Pick something other than $0$ or $1$, since those hide many mistakes.

To prove equivalence, turn one expression into the other using the properties. That covers every input at once. Each step keeps the value the same, by definition.

Worked examples

Common mistakes

Practice problems