Getting It Right: Order of Operations
Order of operations defines the conventional sequence — Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right), known as PEMDAS or BODMAS — for evaluating mathematical expressions with multiple operations. Without these rules, the same expression could produce different answers, so mastering this sequence ensures every calculation gives a single, correct result.
Components of Order of Operations
This section breaks down the steps in the evaluation sequence:
- Parentheses (Brackets): Evaluate expressions inside parentheses or brackets first, working from innermost to outermost.
- Exponents (Orders): Calculate powers and roots next, such as squares, cubes, and square roots.
- Multiplication & Division: Perform these operations from left to right — neither takes priority over the other.
- Addition & Subtraction: Perform these operations last, also from left to right.
Examples of Order of Operations
Parentheses Examples
- Evaluate 3 × (2 + 5): First add inside the parentheses, 2 + 5 = 7, then multiply 3 × 7 = 21.
- Evaluate (8 - 3) × 4: First subtract 8 - 3 = 5, then multiply 5 × 4 = 20.
- Evaluate 2 × (6 + 3) - 1: First 6 + 3 = 9, then 2 × 9 = 18, then 18 - 1 = 17.
Exponents Examples
- Evaluate 3² + 4: First calculate 3² = 9, then add 9 + 4 = 13.
- Evaluate 2 × 5²: First calculate 5² = 25, then multiply 2 × 25 = 50.
- Evaluate (2 + 1)³: First add 2 + 1 = 3, then cube 3³ = 27.
Multiplication & Division Examples
- Evaluate 8 ÷ 2 × 3: Work left to right, 8 ÷ 2 = 4, then 4 × 3 = 12.
- Evaluate 12 × 2 ÷ 6: Work left to right, 12 × 2 = 24, then 24 ÷ 6 = 4.
- A store sells 3 packs of 4 pencils at 2 dollars each, calculated as 3 × 4 × 2 = 24 dollars total.
Full Expression Examples
- Evaluate 5 + 3 × 2: Multiply first, 3 × 2 = 6, then add 5 + 6 = 11 (not 16).
- Evaluate 10 - 2² + 3: Exponent first, 2² = 4, then 10 - 4 + 3 = 9.
- Evaluate 4 × (6 - 2)² ÷ 8: Parentheses first, 6 - 2 = 4, then exponent 4² = 16, then 4 × 16 = 64, then 64 ÷ 8 = 8.