Practice Associativity 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⋆b)⋆c=a⋆(b⋆c)(a \star b) \star c = a \star (b \star c).

(2+3)+4=2+(3+4)(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⋆b=a+b1+aba \star b = \dfrac{a + b}{1 + ab} (relativistic velocity addition for c=1c = 1). Show ⋆\star is associative by computing (a⋆b)⋆c(a \star b) \star c and a⋆(b⋆c)a \star (b \star c) for a=b=c=12a = b = c = \tfrac{1}{2}.

Example 2

medium
Is the operation a⋆b=a−ba\star b = a-b associative? Test 9,4,19,4,1.

Example 3

challenge
Is a⋆b=ab+a+ba\star b = ab+a+b associative? Check (1⋆2)⋆3(1\star 2)\star 3 vs 1⋆(2⋆3)1\star(2\star 3).

Example 4

medium
Find xx: (x+2)+6=x+(2+6)(x+2)+6 = x+(2+6) — what is the value of (2+6)(2+6) here?

Example 5

easy
Which property is illustrated by (a×b)×c=a×(b×c)(a \times b) \times c = a \times (b \times c)?

Example 6

easy
True or false: (20÷5)÷2=20÷(5÷2)(20 \div 5) \div 2 = 20 \div (5 \div 2).

Example 7

easy
Compute (6+9)+1(6 + 9) + 1 and 6+(9+1)6 + (9 + 1). Do they match?

Example 8

easy
Which grouping of 2×5×92\times 5\times 9 is easiest, and what is the product?

Example 9

easy
True or false: (10−4)−2=10−(4−2)(10-4)-2 = 10-(4-2).

Example 10

medium
Use associativity to compute (50×7)×2(50 \times 7) \times 2.

Example 11

easy
Is (8×5)×2=8×(5×2)(8\times 5)\times 2 = 8\times(5\times 2)? Which property?

Example 12

challenge
For a⋆b=2aba\star b=2ab, verify associativity and find (2⋆1)⋆3(2\star 1)\star 3.

Example 13

hard
Define a⋆b=aba \star b = a^b (exponentiation). Show ⋆\star is NOT associative.

Example 14

easy
Compute (2×5)×3(2 \times 5) \times 3 and 2×(5×3)2 \times (5 \times 3).

Example 15

easy
Compute the easier grouping: (17+3)+8(17+3)+8 or 17+(3+8)17+(3+8)?

Example 16

easy
Compute 5+(15+8)5 + (15 + 8) using the easiest grouping.

Example 17

hard
True or false: associativity of ++ implies a+b+c+da + 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)(8+12)+(5+5) using a smart grouping.

Example 20

easy
Calculate (8+3)+7(8 + 3) + 7 and 8+(3+7)8 + (3 + 7). Which is easier? Why?
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