0.4Properties of Operations
Master the commutative, associative, distributive, identity, and inverse properties — the rules that let you rearrange and simplify any algebraic expression.
Why This Matters
Properties of operations are the rules that make algebra flexible and efficient. You'll use the distributive property constantly when expanding expressions, and the commutative and associative properties when simplifying equations in every math course ahead.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do properties of operations let us rewrite expressions in equivalent forms to make calculations easier?
Lesson Overview
Properties of operations are rules that are always true for real numbers. They allow us to rewrite expressions in equivalent forms — which is essential for simplifying and solving equations. The five key properties are: Commutative (order doesn't matter for + and ×), Associative (grouping doesn't matter for + and ×), Distributive (multiply across a sum or difference), Identity (adding 0 or multiplying by 1 leaves a number unchanged), and Inverse (adding the opposite or multiplying by the reciprocal gives the identity element).
Worked Examples
Name the property: 5 + 3 = 3 + 5
The order of the addends changed.
This is the Commutative Property of Addition.
Use the Distributive Property to expand: 4(x + 3)
Multiply 4 by each term inside: 4 · x + 4 · 3
= 4x + 12
Use the Distributive Property to expand: −2(3x − 5)
Multiply −2 by each term: −2 · 3x + (−2)(−5)
= −6x + 10
Simplify using properties: 7 + (x + 3)
Associative Property: (7 + x) + 3 or rearrange
Commutative: x + 7 + 3
Combine constants: x + 10
Simplify: 3(2x + 4) − 5x
Distribute: 6x + 12 − 5x
Combine like terms: (6x − 5x) + 12
= x + 12
Guided Practice
Answers are in the Answer Key section.
Guided Practice Video: Properties of Operations
Watch the guided practice walkthrough for properties of operations, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Name the property: (2 × 5) × 3 = 2 × (5 × 3)
Hint: The numbers are the same but the grouping changed.
Use the Distributive Property to expand: 3(x − 7)
Hint: Multiply 3 by x, then multiply 3 by −7.
Simplify: 5x + 3 + 2x − 1
Hint: Combine the x-terms together and the constants together.
Simplify: 2(3x + 1) + 4x
Hint: Distribute first, then combine like terms.
What is the additive inverse of −8? What is the multiplicative inverse of 3?
Hint: Additive inverse: what adds to give 0? Multiplicative inverse: what multiplies to give 1?
Key Vocabulary
Commutative Property
Changing the order of addends or factors does not change the result.
Example: a + b = b + a and a × b = b × a
Associative Property
Changing the grouping of addends or factors does not change the result.
Example: (a + b) + c = a + (b + c)
Distributive Property
Multiplying a factor by a sum equals the sum of the products.
Example: a(b + c) = ab + ac
Identity Property
Adding 0 or multiplying by 1 leaves a number unchanged.
Example: a + 0 = a and a × 1 = a
Inverse Property
Every number has an additive inverse (opposite) and a multiplicative inverse (reciprocal).
Example: a + (−a) = 0 and a × (1/a) = 1
Like Terms
Terms with the same variable raised to the same power.
Example: 3x and 5x are like terms; 3x and 3x² are not.
Practice Questions
Interactive Practice — 5 Questions
Which property is illustrated by: 5 + 3 = 3 + 5?
Use the Distributive Property to expand: 4(x + 3)
Use the Distributive Property to expand: −2(3x − 5)
Simplify: 3(2x + 4) − 5x
Which property is shown: (2 × 5) × 3 = 2 × (5 × 3)?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Name the property: 4 × 7 = 7 × 4
Expand using the Distributive Property: 5(x + 6)
Expand using the Distributive Property: −3(2x − 4)
Simplify: 4(x + 2) − 3x
Simplify: 2(3x − 1) + 5x
Common Mistakes
Applying the Distributive Property to only the first term — e.g., 3(x + 5) = 3x + 5.
Distribute to every term inside: 3(x + 5) = 3x + 15.
Confusing the Commutative and Associative Properties — thinking they're the same thing.
Commutative changes order (a + b = b + a). Associative changes grouping ((a + b) + c = a + (b + c)).
Forgetting that subtraction and division are NOT commutative — e.g., assuming 8 − 3 = 3 − 8.
Only addition and multiplication are commutative. 8 − 3 = 5, but 3 − 8 = −5.
Distributing a negative sign to only the first term inside parentheses — e.g., −(x + 4) = −x + 4.
The negative distributes to every term: −(x + 4) = −x − 4.
Math Tips
Use the Distributive Property to compute mentally: 8 × 99 = 8(100 − 1) = 800 − 8 = 792. This trick works on any multiplication near a round number.
When combining like terms, underline or circle matching terms first. This prevents accidentally combining x-terms with constants.
The Distributive Property applies to subtraction too: a(b − c) = ab − ac. The sign of each term inside the parentheses must be distributed.
SAT/ACT tip: "which property justifies this step?" questions appear regularly. Know the name and example for each of the five properties.