Order of Operations

Order of Operations

Order of operations gives an expression one consistent value. Grouping symbols come first, then exponents, then multiplication and division, then addition and subtraction.

How to solve these problems

Multiplication and division share a priority and are done left to right. Addition and subtraction also share a priority and are done left to right. A fraction bar groups its whole numerator and denominator.

Worked Example 1: 3 + 4 × 2

3 + 4 × 2

Solution: Multiply first: 4 × 2 = 8. Then 3 + 8 = 11.

Why it works: Multiplication has priority over addition.

Worked Example 2: (3 + 4) × 2

(3 + 4) × 2

Solution: Parentheses first: 3 + 4 = 7. Then 7 × 2 = 14.

Why it works: Grouping changes which operation is performed first.

Worked Example 3: 18 ÷ 3 × 2

18 ÷ 3 × 2

Solution: Work left to right: 18 ÷ 3 = 6; 6 × 2 = 12.

Why it works: Division and multiplication have equal priority.

Worked Example 4: 10 − 4 + 2

10 − 4 + 2

Solution: Work left to right: 10 − 4 = 6; 6 + 2 = 8.

Why it works: Addition does not automatically come before subtraction.

Worked Example 5: 2 + 3² × 4

2 + 3² × 4

Solution: 3² = 9; 9 × 4 = 36; 2 + 36 = 38.

Why it works: Exponents are evaluated before multiplication.

Worked Example 6: 20 − [6 + 2 × 3]

20 − [6 + 2 × 3]

Solution: Inside the brackets, 2 × 3 = 6, then 6 + 6 = 12. Finally 20 − 12 = 8.

Why it works: Finish all operations inside a group using the same priority rules.

Worked Example 7: (8 + 4)/(5 − 2)

(8 + 4)/(5 − 2)

Solution: Numerator: 8 + 4 = 12. Denominator: 5 − 2 = 3. Divide: 12 ÷ 3 = 4.

Why it works: The fraction bar groups both expressions.

Worked Example 8: −3² and (−3)²

−3² and (−3)²

Solution: −3² = −(3 × 3) = −9. (−3)² = (−3) × (−3) = 9.

Why it works: Without parentheses, the exponent acts on 3 before the leading negative is applied.

Worked Example 9: 24 ÷ (2 × 3) + 1

24 ÷ (2 × 3) + 1

Solution: Parentheses: 2 × 3 = 6. Division: 24 ÷ 6 = 4. Addition: 4 + 1 = 5.

Why it works: An explicit group takes priority over the usual left-to-right calculation outside it.

Worked Example 10: 5 + 2[7 − (3 + 1)]

5 + 2[7 − (3 + 1)]

Solution: Inner parentheses: 4. Bracket: 7 − 4 = 3. Multiply: 2 × 3 = 6. Add: 5 + 6 = 11.

Why it works: Nested groups are evaluated from the inside outward; adjacency means multiplication.

Try it again

Cover the solutions and solve the ten examples on your own. Explain which rule you used, then compare your method and answer with the worked solution. Check signs, equivalent forms, and units.