Order of Operations (PEMDAS)

Why is 2 + 3 × 4 equal to 14 and not 20? Because math has an agreed order of operations — a sequence everyone follows so expressions have exactly one correct value.

The rules

1. Parentheses — anything inside brackets first
2. Exponents — powers and roots
3. Multiplication and Division — left to right, equally ranked
4. Addition and Subtraction — left to right, equally ranked

A common mnemonic is PEMDAS (“Please Excuse My Dear Aunt Sally”). The crucial subtlety: multiplication does not “beat” division, and addition does not “beat” subtraction — each pair is worked left to right.

Worked examples

Example 1: 2 + 3 × 4

Multiplication before addition: 2 + 12 = 14

Example 2: 20 ÷ 4 × 2

Equal rank, so left to right: 20 ÷ 4 = 5, then 5 × 2 = 10 (not 20 ÷ 8 = 2.5)

Example 3: 3 + 2² × (8 − 6)

• Parentheses: (8 − 6) = 2
• Exponent: 2² = 4
• Multiplication: 4 × 2 = 8
• Addition: 3 + 8 = 11

Example 4: −3² vs (−3)²

−3² = −(3²) = −9, but (−3)² = 9. The exponent applies only to the 3 unless parentheses include the minus sign. This one trips up calculators and students alike.

Exponents and negatives

When an expression like 2 × 3² appears, the exponent binds to the 3 first: 2 × 9 = 18. To square the whole product, write parentheses: (2 × 3)² = 36.

Why an order at all?

Without a fixed convention, 2 + 3 × 4 could be 14 or 20 depending on the reader — mathematics would be ambiguous. The chosen order is not “natural”; it is a shared agreement that makes notation compact and communication exact. Calculators, spreadsheets, and programming languages all implement the same convention (with minor dialect differences, e.g. implicit multiplication in some calculators).

Quick self-check

Evaluate: 6 + 2 × (5 − 3)² ÷ 4

Answer: parentheses → 2, exponent → 4, multiply → 8, divide → 2, add → 8. If you got 8, you have the order down.