PEMDASstands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction — the order you’re supposed to work through an expression with more than one operation in it. Without an agreed-upon order, 2 + 3 × 4would be ambiguous: is it (2+3)×4 = 20, or 2+(3×4) = 14? PEMDAS exists so everyone gets the same answer, and the agreed answer here is 14 — multiplication happens before addition.
The trap inside the letters: it’s not really six separate steps
The single most common PEMDAS mistake is treating “Multiplication, Division” and “Addition, Subtraction” as four separate steps done strictly in that left-to-right letter order. They’re not. Multiplication and division are the same priority level and you do whichever comes first reading left to right — same for addition and subtraction. In 20 ÷ 4 × 5, doing multiplication before division because M comes before D in PEMDAS gives the wrong path; you go left to right instead: 20 ÷ 4 = 5, then 5 × 5 = 25.
Working through one, step by step
Solve 3 + 2(8 − 5)2.
- Parentheses first: 8 − 5 = 3, giving 3 + 2(3)2.
- Exponents next: 32 = 9, giving 3 + 2(9).
- Multiplication: 2 × 9 = 18, giving 3 + 18.
- Addition: 3 + 18 = 21.
Each step removes exactly one operation and leaves a simpler expression behind — that’s the whole strategy, repeated until one number is left.
The second trap: the invisible negative
What does −22 equal? Not 4. The expression actually means −1 × 22— there’s no parentheses around the −2, so the exponent only applies to the 2, not to the negative sign. Exponents come before multiplication, so you square the 2 first to get 4, then multiply by −1, giving −4. To get a positive 4, the problem would need to say (−2)2, with the negative sign locked inside the parentheses before squaring.
The mistake I see most
Treating PEMDAS as six strictly ordered steps instead of four priority levels, which leads directly to the multiplication-before-division error above. The fix is remembering it as four levels — parentheses, exponents, (multiplication and division together, left to right), (addition and subtraction together, left to right) — not six letters in a strict line.
Want to work through it together?
If you know the PEMDAS song but still get tripped up on problems like the ones above, that’s completely normal — the traps are subtle by design. A little focused practice naming which level you’re on fixes it quickly.
