Associativity Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Associativity.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A property where changing the grouping of operands does not change the result: $(a \star b) \star c = a \star (b \star c)$ .

$(2 + 3) + 4 = 2 + (3 + 4)$ . How you group the additions doesn't matter.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Associativity says changing which operands you group first does not change a sum or product.

Common stuck point: The procedure for associativity is the easy part; the trap is regrouping a subtraction. Asking "Can I move the parentheses among these $+$ or $\times$ operands without changing the result?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can I move the parentheses among these $+$ or $\times$ operands without changing the result?

Worked Examples

Example 1

easy

Show that

(2 + 5) + 4 = 2 + (5 + 4)

by calculating both sides.

Answer

Both equal 11

First step

Left side:

(2 + 5) + 4 = 7 + 4 = 11

Full solution

2
Right side: $2 + (5 + 4) = 2 + 9 = 11$ .
3
Both sides equal 11.
4
The grouping does not change the sum.

The associative property:

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

. You can change the grouping without changing the result.

Example 2

medium

Use associativity to make this multiplication easier:

5 \times 4 \times 6

Example 3

medium

Use associativity to compute

25 \times (4 \times 13)

quickly.

Example 4

medium

Show that subtraction is NOT associative using a specific counterexample.

Example 5

medium

Use associativity (and commutativity) to compute

1 + 2 + 3 + \dots + 10

by pairing.

Example 6

hard

Is exponentiation associative? Test with

2^{(3^2)}

(2^3)^2

Example 7

hard

Define

a \star b = a^b

(exponentiation). Show

\star

is NOT associative.

Example 8

hard

Define

a \diamond b = a - b

. Is

\diamond

associative? Justify.

Example 9

challenge

Define

a \star b = \dfrac{a + b}{1 + ab}

(relativistic velocity addition for

c = 1

). Show

\star

is associative by computing

(a \star b) \star c

and

a \star (b \star c)

for

a = b = c = \tfrac{1}{2}

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Calculate

(8 + 3) + 7

and

8 + (3 + 7)

. Which is easier? Why?

Example 2

medium

Use associativity to simplify:

2 \times 7 \times 5

Example 3

easy

Which property says

(2+3)+4 = 2+(3+4)

Example 4

easy

Compute the easier grouping:

(17+3)+8

17+(3+8)

Example 5

easy

(8\times 5)\times 2 = 8\times(5\times 2)

? Which property?

Example 6

easy

True or false:

(10-4)-2 = 10-(4-2)

Example 7

easy

Regroup to compute

4\times(25\times 7)

easily.

Example 8

easy

Fill in:

(a+b)+c = a+(\underline{\ \ }+c)

Example 9

easy

(12\div 6)\div 2 = 12\div(6\div 2)

Example 10

easy

Which grouping of

2\times 5\times 9

is easiest, and what is the product?

Example 11

medium

Which property is used:

3+(7+x) = (3+7)+x

, and simplify.

Example 12

medium

Use associativity and commutativity to compute

4\times 13\times 25

Example 13

medium

Simplify

(2x+3)+(5)

using associativity of addition.

Example 14

medium

Is the operation

a\star b = a-b

associative? Test

9,4,1

Example 15

medium

Find

x

(x+2)+6 = x+(2+6)

— what is the value of

(2+6)

here?

Example 16

medium

Why can we write

a+b+c

without parentheses?

Example 17

medium

Compute

(50\times 2)\times(5\times 4)

choosing smart regrouping.

Example 18

medium

A student claims

(16-9)-3 = 16-(9-3)

. True or false, and the two values?

Example 19

challenge

a\star b = ab+a+b

associative? Check

(1\star 2)\star 3

1\star(2\star 3)

Example 20

challenge

For

a\star b=2ab

, verify associativity and find

(2\star 1)\star 3

Example 21

challenge

Show subtraction fails associativity in general by finding the difference between the two groupings of

a-b-c

Example 22

medium

Compute

(8+12)+(5+5)

using a smart grouping.

Example 23

easy

Compute

(6 + 9) + 1

and

6 + (9 + 1)

. Do they match?

Example 24

easy

Compute

(2 \times 5) \times 3

and

2 \times (5 \times 3)

Example 25

easy

Use associativity to choose the easier grouping:

(48 + 2) + 17

48 + (2 + 17)

Example 26

easy

True or false:

(20 \div 5) \div 2 = 20 \div (5 \div 2)

Example 27

easy

Compute

5 + (15 + 8)

using the easiest grouping.

Example 28

medium

Use associativity to compute

99 + (1 + 56)

Example 29

medium

Group

2 \times (5 \times 17)

to compute easily.

Example 30

medium

Use associativity to compute

(50 \times 7) \times 2

Example 31

medium

Compute

(0.25 \times 4) \times 99

using associativity.

Example 32

hard

Use associativity and commutativity to compute

4 \times 25 \times 7 \times 5 \times 2

Example 33

hard

Use associativity to compute

\tfrac{1}{2} + \tfrac{1}{3} + \tfrac{1}{2} + \tfrac{2}{3}

Example 34

hard

Use associativity to compute

(125 \times 8) \times 37

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Related Concepts

Addition Multiplication Commutativity Algebraic Representation

Background Knowledge

These ideas may be useful before you work through the harder examples.

additionmultiplication