Order of Operations

An expression like 3 + 4 × 5 could mean two different things: add first and get 35, or multiply first and get 23. To make sure everyone gets the same answer, mathematicians agreed on a ranking called the order of operations: grouping first, then exponents, then multiplication and division, then addition and subtraction. Operations on the same level go from left to right.

Why we need an agreement

Try 3 + 4 × 5. If you work strictly left to right, you get 7 × 5 = 35. If you multiply first, you get 3 + 20 = 23. Both feel natural, so we need a rule that picks one.

The agreed answer is 23. The rule is a convention: an agreement people made so that textbooks, teachers and classmates all match. It is not a law of nature, but nearly all math writing and scientific calculators follow it, so you can rely on it.

The four levels

Work from the top level down:

1. Grouping: parentheses ( ), brackets [ ], braces { }, absolute value bars, and a fraction bar (it groups the top and the bottom). Do what is inside first, starting with the innermost group.
2. Exponents: evaluate powers such as 4².
3. Multiplication and division: these two share one level.
4. Addition and subtraction: these two share one level.

Inside a group the same order applies, so a group is like a small problem of its own.

Ties go left to right

Multiplication and division are on the same level. So are addition and subtraction. When two operations from the same level meet, work them from left to right, the way you read.

In 100 ÷ 10 × 5, go left to right: 100 ÷ 10 = 10, then 10 × 5 = 50. (Multiplying first would give 100 ÷ 50 = 2, which is the value of 100 ÷ (10 × 5), a different expression.)

In 50 − 20 + 10, go left to right: 50 − 20 = 30, then 30 + 10 = 40. (Adding first would give 50 − 30 = 20, which is the answer to 50 − (20 + 10).)

This makes sense because subtraction is adding the opposite: 50 − 20 + 10 means 50 + (−20) + 10, and the minus sign stays attached to the 20.

Grouping symbols

Grouping symbols override the normal ranking: what is inside gets finished first, as one package. (2 + 3) × 4 is 5 × 4 = 20, while 2 + 3 × 4 is 2 + 12 = 14. Same numbers, different meaning.

• Nested groups: start at the innermost pair and work outward. In 2 × (3 + (10 − 4)), do 10 − 4 = 6, then 3 + 6 = 9, then 2 × 9 = 18.
• Fraction bar: the top and the bottom are each worked out first, then divided. The fraction with top 20 − 8 and bottom 2 + 4 means (20 − 8) ÷ (2 + 4) = 12 ÷ 6 = 2.
• Absolute value bars: work out what is inside, then take the absolute value: |2 − 9| + 1 = |−7| + 1 = 7 + 1 = 8.
• Hidden multiplication: 3(4 + 2) means 3 × (4 + 2).
• Division by zero: if a group works out to 0 and you divide by it, as in 9 ÷ (4 − 4), the expression is undefined.

Exponents touch only what is next to them

An exponent applies only to the number (or group) right before it. In 2 × 3², only the 3 is squared: 2 × 9 = 18. To square the whole product you need parentheses: (2 × 3)² = 36.

Negatives need care. In −5² the exponent touches only the 5, so you square 5 first and then apply the minus sign: −(5²) = −25. With parentheses, (−5)² = (−5)(−5) = 25. When you type into a calculator or spreadsheet, use parentheses to show what you mean: (−5)² for 25, and −(5²) for −25. (Some spreadsheets read −5² typed without parentheses as 25, so do not leave it to chance.)

Plugging in a variable

To evaluate an expression such as x² − 4x + 1 for a value of x, replace each x with the value in parentheses, then follow the order. The parentheses matter most when the value is negative, because they keep the exponent and the minus sign from tangling. The last worked example shows it.