Algebra

Order of Operations Explained

The order of operations is a parsing convention. Grouping symbols come first, followed by exponents, multiplication and division from left to right, then addition and subtraction from left to right.

Reviewed 2026-07-26

A hierarchy with equal-priority pairs

PEMDAS is a memory aid, not six independent priority levels. Multiplication and division have equal priority and are handled from left to right. Addition and subtraction also share a level.

That distinction matters in 24 / 6 x 2. Division and multiplication are evaluated from left to right, giving 4 x 2 = 8.

Grouping symbols
Exponents and roots
Multiplication and division from left to right
Addition and subtraction from left to right

Grouping is more than parentheses

Fraction bars, radical bars, absolute-value bars, and function arguments group expressions. In (a+b)/(c+d), both the numerator and denominator are complete grouped expressions.

Nested groups are evaluated from the inside outward. Clear parentheses are preferable when plain-text notation could be read more than one way.

Unary minus and powers

Under standard convention, exponentiation binds before a leading negative sign. Therefore -3^2 means -(3^2) = -9, while (-3)^2 = 9.

Calculator interfaces may display grouping visually, but typed expressions should still include parentheses when the negative number is the base.

Formula-a^2 = -(a^2), while (-a)^2 includes the sign in the base

Common mistakes

Doing multiplication before division regardless of which appears first.
Treating a fraction bar as though it grouped only the nearest number.
Assuming -x^2 and (-x)^2 mean the same expression.

Worked examples

Evaluate 3 + 4 x 2^3

Powers precede multiplication, which precedes addition.

2^3 = 8
4 x 8 = 32
3 + 32 = 35

Result: 35

Evaluate (18 - 6) / 3 + 2

Resolve the numerator before division.

18 - 6 = 12
12 / 3 = 4
4 + 2 = 6

Result: 6

Where this idea is used

Entering unambiguous expressions into a scientific calculator.
Checking algebraic substitutions and spreadsheet formulas.
Reading formulas with fractions, powers, roots, and functions.

Questions about order of operations

Is PEMDAS different from BODMAS?

They describe the same hierarchy; the words differ by region, and multiplication/division still share priority.

Why go left to right?

It supplies a consistent rule for operations at the same precedence level.

Does a calculator always follow this convention?

Standard scientific calculators do, but explicit grouping is the safest way to communicate intent.