A.3 Order of Operations

Equations and formulas often include a number of mathematical operations combining addition, subtraction, multiplication, and division. Some also include exponents and square roots. In complex equations with more than one operation, it is important to perform the operations in a specific sequence. Deviating from this sequence can produce a wrong answer. Table A.2 lists the order of operations for quick reference.

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ORDER OF OPERATIONS: Worked Examples

1. − 3 + 6(4) − 7 = Multiplication
− 3 + 24 − 7 = Addition
21 − 7 = Subtraction
14 = 14 Answer
2. 2(8) + 6 / 3 × 8 = Multiplication
16 + 6 / 3 × 8 = Division
16 + 2 × 8 = Multiplication
16 + 16 = Addition
32 = 32 Answer
3. 32 + 6 / 3 − 12(2) = Square (raise exponent)
9 + 6 / 3 − 12(2) = Division
9 + 2 − 12(2) = Multiplication
9 + 2 − 24 = Addition
11 − 24 = Subtraction
− 13 = − 13 Answer
4. (10 + 6) − 62 / 4 + 3(10) = Within parentheses
16 − 62 / 4 + 3(10) = Square (raise exponent)
16 − 36 / 4 + 3(10) = Division
16 − 9 + 3(10) = Multiplication
16 − 9 + 30 = Subtraction
7 + 30 = Addition
37 = 37 Answer
5. 8 + ( − 4) + 3(12 − 8) = Within parentheses
8 + ( − 4) + 3(4) = Multiplication
8 + (−4) + 12 = Addition
4 + 12 = Addition
16 = 16 Answer

SELF-QUIZ #2: Order of Operations

(Answers to this quiz can be found on page A-6.)

  1. 3(7) − 12 / 3 + 2 =

  2. 4 / 2 + 6 − 2(3) =

  3. − 5(4) + 16 =

  4. 8 + ( − 16) / 4 =

  5. 6 − 3 + 5 − 3(5) + 10 =

  6. 42 / 8 − 4(3) + (8 − 3) =

  7. (14 − 6) + 72 / 9 + 4 =

  8. (54 − 18) / 4 + 7 × 3 =

  9. 32 − 4(3 + 4) + 8 =

  10. 100 × 3 − 87 =