Order of operations

Why order matters#

Consider 8 + 2 × 3. Adding first gives 30, but multiplying first gives 14. Standard order makes 14 the agreed result:

1. Multiply: 2 × 3 = 6.
2. Add: 8 + 6 = 14.

A useful sequence is:

1. Grouping symbols
2. Exponents
3. Multiplication and division, from left to right
4. Addition and subtraction, from left to right

Grouping symbols#

Parentheses tell you to evaluate an inner expression first:

(8 + 2) × 3 = 10 × 3 = 30

Nested groups work from the inside outward:

2 × [3 + (7 - 4)] = 2 × [3 + 3] = 12

A fraction bar also groups its numerator and denominator. In (6 + 2) / (5 - 1), calculate both grouped parts before dividing: 8 / 4 = 2.

Exponents#

An exponent is repeated multiplication. In 3 + 2² × 5, calculate 2² first:

1. 2² = 4
2. 4 × 5 = 20
3. 3 + 20 = 23

The exponent applies only to its stated base. -3² is normally interpreted as -(3²) = -9, while (-3)² = 9 because the parentheses make -3 the whole base.

Equal priority means left to right#

Multiplication does not always come before division. They share a level:

24 ÷ 6 × 2 = 4 × 2 = 8

Reading it as 24 ÷ 12 would incorrectly perform the later multiplication first.

Addition and subtraction work the same way:

15 - 6 + 2 = 9 + 2 = 11

Check the result#

Write one change per line for a long expression. Keep untouched parts visible, estimate the likely size, and use a calculator only after you know what expression should be entered.

A common mistake is treating a memorized acronym as permission to ignore left-to-right order. The priority groups matter more than the initials: multiplication and division are partners, as are addition and subtraction.