Practice Associative Property in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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}

Example 2

medium

Is the operation

a\star b = a-b

associative? Test

9,4,1

Example 3

challenge

a\star b = ab+a+b

associative? Check

(1\star 2)\star 3

1\star(2\star 3)

Example 4

medium

Find

x

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

— what is the value of

(2+6)

here?

Example 5

easy

Which property is illustrated by

(a \times b) \times c = a \times (b \times c)

Example 6

easy

True or false:

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

Example 7

easy

Compute

(6 + 9) + 1

and

6 + (9 + 1)

. Do they match?

Example 8

easy

Which grouping of

2\times 5\times 9

is easiest, and what is the product?

Example 9

easy

True or false:

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

Example 10

medium

Use associativity to compute

(50 \times 7) \times 2

Example 11

easy

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

? Which property?

Example 12

challenge

For

a\star b=2ab

, verify associativity and find

(2\star 1)\star 3

Example 13

hard

Define

a \star b = a^b

(exponentiation). Show

\star

is NOT associative.

Example 14

easy

Compute

(2 \times 5) \times 3

and

2 \times (5 \times 3)

Example 15

easy

Compute the easier grouping:

(17+3)+8

17+(3+8)

Example 16

easy

Compute

5 + (15 + 8)

using the easiest grouping.

Example 17

hard

True or false: associativity of

+

implies

a + b + c + d

has a unique value regardless of grouping.

Example 18

medium

Show that subtraction is NOT associative using a specific counterexample.

Example 19

medium

Compute

(8+12)+(5+5)

using a smart grouping.

Example 20

easy

Calculate

(8 + 3) + 7

and

8 + (3 + 7)

. Which is easier? Why?

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Related Concepts

Addition Multiplication Commutative Property Algebraic Representation