Order of Operations

The order

Work through an expression in this order. First, parentheses — compute whatever is inside grouping symbols. Second, exponents — evaluate powers like $2^{3}$. Third, multiplication and division, working left to right. Fourth, addition and subtraction, working left to right.

Notice that's four levels, not six. Multiplication and division share one level, and addition and subtraction share one level. That detail is where most mistakes come from, so it gets its own section below.

Grouping symbols include more than parentheses: a fraction bar groups everything above it and everything below it. In $\dfrac{8 + 4}{3}$, you compute $8 + 4 = 12$ first, then divide to get $4$.

The left-to-right rule

PEMDAS does not mean multiplication always beats division. They are the same rank, so you take them in the order they appear, left to right. In $24 \div 6 \cdot 2$, the division comes first: $24 \div 6 = 4$, then $4 \cdot 2 = 8$. Multiplying first would give $24 \div 12 = 2$, which is wrong.

The same goes for addition and subtraction. In $20 - 8 + 3$, subtract first because it appears first: $20 - 8 = 12$, then $12 + 3 = 15$. Adding first would give $20 - 11 = 9$, which is wrong.

Parentheses override everything

Parentheses are how you force a different order. $3 + 4 \cdot 5 = 23$, but $(3 + 4) \cdot 5 = 35$ — the parentheses make the addition happen first. If parentheses are nested inside other parentheses, work from the innermost pair outward.

This is also why parentheses matter with exponents and negatives: $(-3)^{2} = 9$ because the parentheses square the whole $-3$, while $-3^{2} = -9$ because the exponent applies only to the $3$.

Worked examples

Try one yourself

Common questions