Order of Operations: PEMDAS, BODMAS and the Mistakes Everyone Makes
What is 8 − 2 × 3? If you said 18 you worked left to right; if you said 2 you followed the order of operations. Mathematicians agree on one order so that every expression has exactly one value. This guide explains the rules, why they exist, and where even confident students slip.
The rule in one line
Work out expressions in this order: brackets (parentheses), then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. In the United States this is taught as PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction); in the UK and elsewhere as BODMAS or BIDMAS (Brackets, Orders or Indices, Division, Multiplication, Addition, Subtraction). They are the same rule.
Why multiplication comes before addition
It is a convention, but a sensible one. Multiplication is shorthand for repeated addition, so 3 × 4 is really "4 + 4 + 4", a single chunk of value. When you write 2 + 3 × 4 you mean "two, plus three fours". Giving multiplication priority means polynomials and everyday formulas can be written without extra brackets: 3x² + 2x + 1 does not need to be written ((3 × (x²)) + (2 × x)) + 1.
The most common mistake: PEMDAS is not a strict ladder
Many people read PEMDAS as six separate steps and always multiply before dividing, or add before subtracting. That is wrong. Multiplication and division share one level, and so do addition and subtraction. Within a level you work from left to right.
Example: 24 ÷ 4 × 3
Division and multiplication are on the same level, so go left to right: 24 ÷ 4 = 6, then 6 × 3 = 18. (Multiplying first would give 24 ÷ 12 = 2, which is wrong.)
Example: 10 − 4 + 3
Left to right: 10 − 4 = 6, then 6 + 3 = 9. (Adding first would give 10 − 7 = 3, which is wrong.)
Working through a full expression
Evaluate 6 + 2 × (9 − 4)² ÷ 5
- Brackets: 9 − 4 = 5, so the expression becomes 6 + 2 × 5² ÷ 5.
- Exponents: 5² = 25, giving 6 + 2 × 25 ÷ 5.
- Multiplication and division, left to right: 2 × 25 = 50, then 50 ÷ 5 = 10, giving 6 + 10.
- Addition: 6 + 10 = 16.
Rewrite the whole expression after every step. It takes a few extra seconds, but it makes it nearly impossible to lose track of which part you have already handled.
Brackets can be implicit
A few notations hide brackets, and ignoring them is another frequent source of error:
- Fraction bars. In (3 + 5) ⁄ (2 + 2), the top and bottom are each treated as if they were in brackets, so the value is 8 ⁄ 4 = 2, not 3 + 5⁄2 + 2.
- Roots. A square-root sign covers everything under its bar: √(9 + 16) = √25 = 5, which is not 3 + 4.
- Exponents on brackets. (−3)² = 9 because the bracket includes the minus sign, but −3² = −9 because the exponent applies only to the 3.
Negative numbers and exponents
That last distinction trips up many calculators and many people. In −3², the exponent has higher priority than the negation, so you square 3 first and then apply the minus sign. Compare: (−3)² = (−3) × (−3) = 9, while −3² = −(3 × 3) = −9. If you ever want the other meaning, add brackets. For more on signs, see our guide to negative numbers.
Ambiguous-looking expressions
An expression such as 8 ÷ 2(2 + 2) causes arguments online. Strictly by the rules, you evaluate the bracket (4), then work left to right: 8 ÷ 2 = 4, then 4 × 4 = 16. Some textbooks, however, treat a number written directly next to a bracket as binding more tightly, which would give 8 ÷ 8 = 1. The lesson is practical: when you write expressions for other people, add brackets to remove any doubt, and when you read one, notice whether the source is following the left-to-right convention.
A quick self-test
- 12 − 3 × 2 = ? (Answer: 6)
- (12 − 3) × 2 = ? (Answer: 18)
- 20 ÷ 5 × 2 = ? (Answer: 8)
- 2 + 3² × 4 = ? (Answer: 38)
- (7 + 5) ÷ (4 − 1) = ? (Answer: 4)
If you missed any, check whether you broke the left-to-right rule within a level. Mixed expressions with brackets, negatives and two operations appear in the Arithmetic Blitz mode, which is a good way to build the habit under time pressure.