The associative property says that when you add or multiply three or more numbers, the way you group them does not change the result: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c). It lets you pick the easiest pair first. It does not work for subtraction or division.
In Spanish
propiedad asociativa
Also called
associative law, grouping property, associative property of addition, associative property of multiplication
With + and ×, you may group any pair first, so choose the pair that makes a ten or a hundred. With − and ÷ the grouping changes the answer.
Formal definition and notation
An operation · is associative on a set if (a · b) · c = a · (b · c) for all a, b, c in the set. For all real numbers, addition and multiplication are associative: (a + b) + c = a + (b + c) and (a · b) · c = a · (b · c). Because of this, expressions like a + b + c and abc need no parentheses.
Symbol
Read it as
Meaning
(a + b) + c = a + (b + c)
a plus b, plus c equals a plus, b plus c
associative property of addition
(ab)c = a(bc)
a b times c equals a times b c
associative property of multiplication
Key facts
Only the parentheses move; the numbers stay in the same order.
Group to make tens: 36 + 18 + 2 = 36 + (18 + 2) = 36 + 20 = 56.
Group to make 100 or 10 when multiplying: 7 × 25 × 4 = 7 × (25 × 4) = 700.
In algebra it simplifies products: 3(4x) = (3 × 4)x = 12x.
Subtraction and division are not associative: (20 − 6) − 4 = 10 but 20 − (6 − 4) = 18.
Distributive mixes two operations: 4(x + 2) = 4x + 8. Associative uses one operation and only regroups.
Quick check: 3 questions
Question 1 of 3
Which equation shows the associative property?
Show answer and explanation
Answer: B. Only choice B keeps the order 6, 2, 5 and moves the parentheses. A is commutative, C is distributive, and D is the identity property.
Question 2 of 3
Which grouping makes 19 + 37 + 3 easiest to add mentally?
Show answer and explanation
Answer: C.37 + 3 = 40, and 19 + 40 = 59. Grouping 19 and 37 first gives a harder sum, and D also reorders the numbers, which uses the commutative property too.
Question 3 of 3
What is (40 ÷ 10) ÷ 2 compared with 40 ÷ (10 ÷ 2)?
Show answer and explanation
Answer: A.(40 ÷ 10) ÷ 2 = 4 ÷ 2 = 2 and 40 ÷ (10 ÷ 2) = 40 ÷ 5 = 8. They differ because division is not associative.
Associative Property: frequently asked questions
What is an example of the associative property of multiplication?
(2 × 3) × 4 = 2 × (3 × 4). On the left, 6 × 4 = 24; on the right, 2 × 12 = 24. Both groupings give 24. Students use this to multiply quickly, for example 5 × 13 × 2 = 13 × (5 × 2) = 130, which also uses the commutative property to bring 5 and 2 together.
Does the associative property work with subtraction?
No. (9 − 4) − 2 = 3, but 9 − (4 − 2) = 7. The fix is to rewrite subtraction as adding the opposite: 9 + (−4) + (−2). Now every operation is addition, and any grouping gives 3. The same trick turns division into multiplying by a reciprocal.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 1.OA.B.3, CCSS 3.OA.B.5, CCSS 6.EE.A.3
Cite this entry
Effortless Math Team. “Associative Property.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/associative-property/
Flashcard
TermAssociative Property(a + b) + c = a + (b + c)
DefinitionRegrouping does not change a sum or product: (3 × 5) × 2 = 3 × (5 × 2) = 30. Not true for subtraction or division.
Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2SubtractionSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2, Grade 3DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6DivisorA divisor is the number you divide by. In 56 ÷ 8 = 7, the divisor is 8: it tells how many equal groups to make, or how big each group is. The divisor can never be 0. A divisor of a whole number is also one of its factors, because it divides that number with no remainder.Operations · Grade 3, Grade 4, Grade 5Commutative PropertyThe commutative property says that changing the order of the numbers does not change the result of adding or multiplying them: a + b = b + a and a × b = b × a. For example, 4 + 9 = 9 + 4 and 3 × 5 = 5 × 3. Subtraction and division are not commutative.Operations · Grade 1, Grade 3, Grade 6RegroupingRegrouping is trading 10 of one place value for 1 of the next larger place, or 1 for 10 of the next smaller place, so that column addition or subtraction works. In 47 + 38, the 15 ones become 1 ten and 5 ones; in 52 − 27, 1 ten becomes 10 ones. It is also called carrying and borrowing.Operations · Grade 1, Grade 2, Grade 3Distributive PropertyThe distributive property says that multiplying a number by a sum gives the same result as multiplying it by each term and then adding: a(b + c) = ab + ac. It multiplies a factor by each term inside parentheses, so 3(x + 4) = 3x + 12. It works with subtraction too, and using it backward is factoring.Algebra · Grade 6, Grade 7, Grade 8ReciprocalThe reciprocal of a number is 1 divided by that number, so the two multiply to 1. For a fraction, swap the numerator and denominator: the reciprocal of 34 is 43, and the reciprocal of 5 is 15. Zero has no reciprocal. Dividing by a fraction is the same as multiplying by its reciprocal.Number System · Grade 5, Grade 6, Grade 7Order of OperationsThe order of operations is the agreed sequence for evaluating an expression: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Without it the same expression would have several answers. Many students remember it as PEMDAS.Algebra · Grade 5, Grade 6, Grade 7Equivalent ExpressionsEquivalent expressions are expressions that have the same value for every value of the variable, even though they look different. For example, 3(x + 2) and 3x + 6 are equivalent: both equal 15 when x = 3 and 0 when x = −2. You build them with the properties of operations, such as the distributive property.Algebra · Grade 6, Grade 7, Algebra 1Combining Like TermsCombining like terms is simplifying an expression by adding or subtracting the terms that have exactly the same variable part. You add their coefficients and keep the variable part: 3x + 2x = 5x. Constants combine with constants. So 3x + 5 + 2x − 1 simplifies to 5x + 4.Algebra · Grade 6, Grade 7, Algebra 1ExpressionAn expression is a number, variable, or combination of numbers and variables joined by operations, without an equal sign. Examples are 7, 4x + 9 and 3(a − 2). Expressions can be simplified, but they are not solved unless they are part of an equation. You evaluate one by replacing each variable with a number.Algebra · Grade 6, Grade 7, Grade 8Identity PropertyThe identity property says a number keeps its value when you add 0 or multiply by 1: a + 0 = a and a × 1 = a. So 0 is the additive identity and 1 is the multiplicative identity. For example, 15 + 0 = 15 and 15 × 1 = 15. It is how equivalent fractions work.Operations · Grade 1, Grade 3, Grade 6