The order of operations is the set of rules that tells you which part of a math expression to calculate first. The most common way to remember it is PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Follow that order and every expression has exactly one correct answer. On this page you will find the PEMDAS rule explained step by step, worked examples with parentheses and exponents, the mistakes students make most, and an interactive quiz to practice right away.
What PEMDAS means: the order of operations
PEMDAS is a mnemonic for the order in which operations are carried out. Many students remember it with the phrase "Please Excuse My Dear Aunt Sally." Work down the ladder from top to bottom:
- P — Parentheses. Do everything inside grouping symbols first: ( ), [ ], { }.
- E — Exponents. Then powers and roots, such as 3² or √9.
- M and D — Multiplication and Division. These share the same priority. Do them together, working left to right.
- A and S — Addition and Subtraction. These also share the same priority. Do them last, again left to right.
The single most important idea is the one most students miss: PEMDAS is not six separate steps. Multiplication does not always come before division, and addition does not always come before subtraction. Each of those pairs is one level, and within a level you simply move from left to right.
The left-to-right rule (M/D and A/S are equal)
Because multiplication and division sit on the same level, you handle them in the order they appear, reading left to right — not "all multiplication, then all division." The same is true for addition and subtraction.
A classic trap is 20 ÷ 4 × 5. If you did multiplication first (4 × 5 = 20), you would get 20 ÷ 20 = 1 — which is wrong. Since division is on the left, you do it first: 20 ÷ 4 = 5, then 5 × 5 = 25. The same logic applies to 10 − 6 + 2: work left to right to get 4 + 2 = 6, not 10 − 8 = 2.
PEMDAS vs BODMAS: the same rule, different name
In the UK, Australia, and many other countries the same rule is taught as BODMAS (sometimes BIDMAS or BEDMAS). The letters look different, but the order is identical — only the vocabulary changes.
- B — Brackets is the same as P — Parentheses.
- O — Orders (or I — Indices, E — Exponents) means powers and roots.
- DM and MD are the same equal-priority pair, done left to right.
- AS is addition and subtraction, done left to right.
So PEMDAS and BODMAS always give the same answer. The comparison below lines them up letter by letter.
| Step | PEMDAS | BODMAS | What you do |
|---|---|---|---|
| 1 | Parentheses | Brackets | Inside grouping symbols first |
| 2 | Exponents | Orders / Indices | Powers and roots |
| 3 | Multiplication / Division | Division / Multiplication | Equal priority, left → right |
| 4 | Addition / Subtraction | Addition / Subtraction | Equal priority, left → right |
Order of operations examples (step by step)
Example 1. Multiplication before addition
Expression: 3 + 4 × 2
Step 1 — Parentheses / Exponents: none here, so move on.
Step 2 — Multiplication / Division: 4 × 2 = 8. The expression is now 3 + 8.
Step 3 — Addition / Subtraction: 3 + 8 = 11.
A common wrong answer is 14, which comes from adding 3 + 4 first. Multiplication has to happen before addition.
Example 2. Parentheses, exponents, then left to right
Expression: (6 + 2) × 3² ÷ 4 − 5
Step 1 — Parentheses: 6 + 2 = 8. Now: 8 × 3² ÷ 4 − 5.
Step 2 — Exponents: 3² = 9. Now: 8 × 9 ÷ 4 − 5.
Step 3 — Multiplication / Division (left to right): 8 × 9 = 72, then 72 ÷ 4 = 18. Now: 18 − 5.
Step 4 — Addition / Subtraction: 18 − 5 = 13.
Example 3. The left-to-right trap
Expression: 20 ÷ 4 × 5
There are no parentheses or exponents, so we go straight to multiplication and division. They are equal priority, so we read left to right.
Step 1: 20 ÷ 4 = 5 (division is leftmost). Now: 5 × 5.
Step 2: 5 × 5 = 25.
If you had multiplied first (4 × 5 = 20), you would get 20 ÷ 20 = 1 — a classic mistake. The letter order M-then-D does not mean multiplication wins; left-to-right does.
Common PEMDAS mistakes
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Thinking multiplication always comes before division (or addition before subtraction) because of the letter order M-D and A-S.
Multiplication and division are equal, and so are addition and subtraction. Within each pair, work left to right. In 20 ÷ 4 × 5 you divide first because it is on the left, giving 25 — not 1.
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Ignoring the left-to-right rule for subtraction, so 10 − 6 + 2 is read as 10 − 8 = 2.
Go left to right: 10 − 6 = 4, then 4 + 2 = 6. Addition does not "jump the line" ahead of the subtraction to its left.
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Skipping parentheses or doing them last because they appear in the middle of the expression.
Grouping symbols are always first, no matter where they sit. In 2 × (3 + 4), you compute 3 + 4 = 7 before multiplying, giving 14 — not 2 × 3 + 4 = 10.
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Misreading the scope of an exponent, treating −3² and (−3)² as the same thing.
An exponent applies only to the number directly beneath it. −3² means −(3²) = −9, while (−3)² means (−3)(−3) = 9. Use parentheses when you want the sign included.
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Working nested parentheses from the outside in.
Start with the innermost grouping and work outward. In 2 × [3 + (4 − 1)], do 4 − 1 = 3 first, then 3 + 3 = 6, then 2 × 6 = 12.
Frequently asked questions
What does PEMDAS stand for?
PEMDAS stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. It is the order in which you evaluate a math expression. Many students remember it with the phrase "Please Excuse My Dear Aunt Sally."
What is the difference between PEMDAS and BODMAS?
There is no difference in the math — only in the names. BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) is taught in the UK and many other countries, while PEMDAS is used in the US. Brackets are the same as parentheses, and Orders (or Indices) are the same as exponents. Both give the identical answer for every expression.
Does multiplication always come before division in PEMDAS?
No. Multiplication and division have equal priority. You do whichever appears first when reading from left to right. For example, in 20 ÷ 4 × 5 you divide first because it is on the left, then multiply, giving 25.
Why do we even need an order of operations?
Without an agreed order, the same expression could give different answers to different people. For instance, 3 + 4 × 2 would be 11 if you multiply first or 14 if you add first. PEMDAS makes sure every expression has exactly one correct result.
How do you handle exponents and negative signs together?
An exponent applies only to the value directly under it. So −3² equals −(3 × 3) = −9, because the exponent binds to the 3 before the negative sign is applied. If you want the negative included, write (−3)², which equals 9. Parentheses control what the exponent covers.
What do you do with nested parentheses or brackets?
Work from the innermost group outward. In 2 × [3 + (4 − 1)], first do 4 − 1 = 3, then 3 + 3 = 6, then 2 × 6 = 12. Each layer of grouping is resolved before the one around it.
