What is the order of operations?

In turn: 1) brackets, 2) powers and roots, 3) multiplication and division, 4) addition and subtraction. Operations of the same level (such as multiplication and division) are done left to right.

Fourteen. Multiplication takes priority over addition, so first compute 3 · 4 = 12, and only then 2 + 12 = 14. The answer 20 (that is 5 · 4) is wrong because it ignores the order of operations.

What are brackets for?

Brackets change the natural order — whatever is inside is computed first. That is why (2 + 3) · 4 = 20, even though without the brackets 2 + 3 · 4 = 14. With nested brackets, start from the innermost.

Is multiplication always done before division?

No. Multiplication and division are on the same level, so they are done left to right. For example in 12 : 3 · 2 you first compute 12 : 3 = 4, then 4 · 2 = 8 — not 3 · 2 first.

Basic level

Order of operations

When several operations meet in one expression, we evaluate them in a fixed order: brackets first, then powers, then multiplication and division, and finally addition and subtraction. Learn the rules and practise them on examples.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

AdditionAddition combines two or more numbers into one — the sum. Learn the names of the parts, the laws of addition (commutativity, associativity, identity element) and how to add in columns with carrying.
SubtractionSubtraction is the inverse of addition — take the subtrahend away from the minuend to get the difference. Learn the names, the properties, the link to addition and how to subtract in columns with borrowing.
MultiplicationMultiplication is repeated addition of the same number. Learn the names of the factors and the product, the laws of multiplication (commutativity, associativity, distributivity), the times tables and column multiplication.
DivisionDivision is the inverse of multiplication — divide the dividend by the divisor to get the quotient. Learn the names, the link to multiplication, division with a remainder and why you must never divide by zero.
PowersA power is shorthand for multiplying the same factor by itself. Learn the base and the exponent, the laws of exponents, powers of a product and of a quotient, zero and negative exponents, the monotonicity of exponentiation, and scientific notation.
RootsA root is the inverse of raising to a power. Learn square and cube roots, arithmetic with roots, taking a factor out of a radical and absorbing one under it, estimating and comparing roots, and rationalising a denominator.

Where this is used

Real situations where you count exactly the way this lesson teaches:

A shopping basket
Three yoghurts at 4.50 and two cheeses at 7.20 come to 3 · 4.50 + 2 · 7.20 = 27.90. Work strictly left to right instead and you get 111.60 — the order is the difference between a bill and nonsense.
Bookkeeping
The gross on an invoice is (12 · 85 + 40) · 1.23 = 1,303.80: line items and delivery first, tax last. Without the bracket the same numbers give 1,069.20.
See the lesson: Percentages
Wholesale
A case of 24 cans costs 43.20 and the customer wants five of them: 43.20 : 24 · 5 = 9. Division and multiplication rank equally, so they run left to right — key the same sum in as 43.20 : (24 · 5) and the calculator answers 0.36, a price nobody could ever sell at.

All formulas

Multiplication before addition
$2 + 3 \cdot 4 = 14$
first 3 · 4 = 12, then 2 + 12
Brackets first
$(2 + 3) \cdot 4 = 20$
brackets change the order: first 2 + 3
Power before multiplication
$2^3 \cdot 2 = 16$
first 2³ = 8, then 8 · 2
Left to right
$20 - 4 - 3 = 13$
operations of the same rank are done left to right

When several different operations meet in one expression, the result depends on the order in which you carry them out. So that everyone gets the same answer, mathematics fixes one standard order.

The order-of-operations rule

Carry out operations in this order:

brackets — whatever is inside first (from the innermost),
powers and roots,
multiplication and division,
addition and subtraction.

Operations on the same level (multiplication and division together, or addition and subtraction together) are done left to right.

Why the order matters

Compare two expressions that differ only by a bracket:

2 + 3 \cdot 4 = 14 \qquad \text{but} \qquad (2 + 3) \cdot 4 = 20

In the first, multiplication takes priority, so compute $3 \cdot 4 = 12$ , then $2 + 12 = 14$ . In the second, the bracket says add first: $2 + 3 = 5$ , then $5 \cdot 4 = 20$ .

Same level — left to right

Multiplication and division rank equally, so you don't "always multiply first" — you go from the left:

12 : 3 \cdot 2 = 4 \cdot 2 = 8

Had we multiplied $3 \cdot 2 = 6$ first, we would get the wrong $12 : 6 = 2$ . The same goes for addition and subtraction: $20 - 4 - 3 = 13$ , working left to right.

Compute: 20 − 4 · 2 + 6 : 3.

The same order applies when a letter stands where a number would. In the last exercise below, type an expression — what counts is its value, not how you write it, so any equivalent form passes.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 7Score: 0

2 + 3 × 4 =

Common mistakes

Working left to right, ignoring priority — $2 + 3 \cdot 4$ is $14$ , not $20$ .
"Multiplication always before division" — no: they rank equally, so go left to right.
Losing brackets — a bracket can change the result entirely, so copy it carefully.

Formula card

Topic: Order of operations

Multiplication before addition
$2 + 3 \cdot 4 = 14$
first 3 · 4 = 12, then 2 + 12
Brackets first
$(2 + 3) \cdot 4 = 20$
brackets change the order: first 2 + 3
Power before multiplication
$2^3 \cdot 2 = 16$
first 2³ = 8, then 8 · 2
Left to right
$20 - 4 - 3 = 13$
operations of the same rank are done left to right

Frequently asked questions

Plus, minus, times and division signs and digits floating in a geometric grid, joined by emerald lines of the order of operations, against a cosmic nebula in violet and magenta

Arithmetic

The grammar of arithmetic — why the order of operations is a convention

Nobody invented +, −, × and ÷ overnight, and the fight over 6 ÷ 2(1+2) still splits the internet. Here is why the notation of arithmetic is a human convention, the results are absolute truth, and division by zero simply cannot be defined.

10 min