What Is the Order of Operations?
Imagine two students evaluate the expression 3 + 4 × 2. One adds first and gets 14; the other multiplies first and gets 11. Without a universally agreed-upon order, the same expression could produce different answers — and math would be chaos.
Some people remember it with the mnemonic "Please Excuse My Dear Aunt Sally."
The key detail that trips up most students is that multiplication and division share the same priority level, and addition and subtraction share the same priority level. Within each level, you simply work from left to right.
This convention is not a law of nature — it is a universally adopted agreement that keeps mathematical communication unambiguous across classrooms, textbooks, and computer programs worldwide.
The PEMDAS Rules Step by Step
Follow these steps in order every time you evaluate an expression.
- Step 1 — Parentheses (and brackets)
Evaluate everything inside parentheses
(), brackets[], or braces{}first. If parentheses are nested, start with the innermost pair and work outward. - Step 2 — Exponents
Evaluate all powers and roots (square roots are the same as raising to the 1/2 power).
- Step 3 — Multiplication and Division
Perform these from left to right. Multiplication does not outrank division — they are equal. Whichever appears first as you scan left to right gets done first. For example, in
12 ÷ 3 × 2, you divide first to get 4, then multiply by 2 to get 8. If you multiplied first you would get a different (wrong) answer. - Step 4 — Addition and Subtraction
Perform these from left to right. Again, addition does not outrank subtraction. In
10 − 4 + 3, subtract first to get 6, then add 3 to get 9.
Remembering that each pair (M/D and A/S) is resolved left to right is the single most important nuance of PEMDAS and the source of most student errors.
Worked Examples
Let us trace through several expressions step by step.
Evaluate 8 + 2 × 5.
Multiplication before addition: 2 × 5 = 10.
Then 8 + 10 = 18.
18Evaluate (3 + 4)² − 10 ÷ 2.
Parentheses first: 3 + 4 = 7.
Exponent: 7² = 49.
Division: 10 ÷ 2 = 5.
Subtraction: 49 − 5 = 44.
44Evaluate 6 ÷ 2(1 + 2). This famous internet debate is resolved by strict PEMDAS.
Parentheses first: 1 + 2 = 3. Now the expression is 6 ÷ 2 × 3.
Multiplication and division are at the same level, so go left to right: 6 ÷ 2 = 3, then 3 × 3 = 9.
9Evaluate 4² − 3 × (6 − 2) + 1.
Parentheses: 6 − 2 = 4.
Exponent: 4² = 16.
Multiplication: 3 × 4 = 12.
Now left to right: 16 − 12 + 1 = 4 + 1 = 5.
5Working through these step by step builds the discipline to avoid skipping steps and making errors on more complex problems.
The Left-to-Right Rule
One of the most persistent misconceptions is that multiplication always comes before division, and addition always comes before subtraction, because of the order of the letters in PEMDAS. This is wrong. The acronym groups M and D together and A and S together — they are pairs at the same level, resolved left to right.
Consider 24 ÷ 6 × 2.
If you incorrectly multiply first: 6 × 2 = 12, then 24 ÷ 12 = 2 — wrong.
Correct left-to-right evaluation: 24 ÷ 6 = 4, then 4 × 2 = 8.
8Now consider 10 − 3 + 2.
Subtracting and adding left to right: 10 − 3 = 7, then 7 + 2 = 9.
If you incorrectly added first: 3 + 2 = 5, then 10 − 5 = 5 — wrong.
9A useful way to avoid this confusion is to think of PEMDAS as having only four levels, not six: (1) Parentheses, (2) Exponents, (3) Multiplication/Division, (4) Addition/Subtraction. Within levels 3 and 4, the tie-breaker is always left to right. Once this clicks, the order of operations becomes almost impossible to misapply.
Common Mistakes Students Make
Beyond the left-to-right confusion, several other errors appear again and again.
In 2(3 + 5)², some students distribute the 2 inside the parentheses to get (6 + 10)². This is incorrect. Evaluate the parentheses first: 3 + 5 = 8, apply the exponent: 8² = 64, then multiply: 2 × 64 = 128.
In 3 × [2 + (4 − 1)²], start with the innermost group: 4 − 1 = 3, then the exponent: 3² = 9, then inside the brackets: 2 + 9 = 11, finally multiply: 3 × 11 = 33. Students who try to rush through nested groups often lose track.
The expression −3² means −(3²) = −9, because exponentiation has higher priority than the negation (which is multiplication by −1). If you want to square negative three, you must write (−3)² = 9. This distinction is tested constantly on standardized exams.
If you ever feel uncertain about how an expression should be evaluated, you can use Solver AI to snap a photo and see each step broken down in the correct order.
PEMDAS vs BODMAS vs BEDMAS
If you have seen the acronyms BODMAS or BEDMAS and wondered whether they represent different rules, the answer is simple: they all describe the exact same order of operations. The differences are purely regional naming conventions.
| Acronym | Region | Letters |
|---|---|---|
| PEMDAS | United States | Parentheses, Exponents, Multiplication, Division, Addition, Subtraction |
| BODMAS | United Kingdom, India, Australia | Brackets, Orders, Division, Multiplication, Addition, Subtraction |
| BEDMAS | Canada | Brackets, Exponents, Division, Multiplication, Addition, Subtraction |
Notice that BODMAS lists Division before Multiplication while PEMDAS lists Multiplication before Division. This causes some students to think the two systems give different answers — they do not. In every system, multiplication and division are treated as equal-priority operations resolved left to right. The letter order within the acronym is just a consequence of needing to spell something pronounceable, not a ranking. Similarly, "Brackets" and "Parentheses" are the same thing, and "Orders" and "Exponents" refer to the same operation (powers and roots).
No matter which acronym you learned, the underlying math is identical everywhere in the world.