ArithmeticArithmetic FoundationsMiddle school

Order of Operations

Evaluate expressions by handling grouping, exponents, multiplication/division and addition/subtraction in the right order.

An expression like $8 - 2 \times 3$ only has one correct value because everyone agrees on the order in which operations are carried out. Without that agreement, the same line of symbols could mean 18 or 2, and calculators, spreadsheets and textbooks would disagree.

The convention is: work inside grouping symbols first, then evaluate powers and roots, then do multiplication and division as they appear from left to right, and finally addition and subtraction from left to right. Multiplication does not outrank division; they share a level, and so do addition and subtraction.

Key terms

Operation precedence: Grouping first, then powers, then multiplication and division left to right, then addition and subtraction left to right.
$2 + 3 \times 4 = 14$

Key ideas

Grouping symbols include parentheses, brackets, fraction bars and the bar of a square root.
Within one level, always move left to right: $12 \div 3 \times 2 = 8$ , not $2$ .
A negative sign in front of a power is applied after the power: $-3^2 = -9$ , while $(-3)^2 = 9$ .
Rewrite step by step instead of doing several operations in your head.

Worked example

Evaluate $20 - 4 \times (5 - 2)^2 \div 6$ .

Parentheses: $5 - 2 = 3$ .
Power: $3^2 = 9$ .
Multiply and divide left to right: $4 \times 9 = 36$ , then $36 \div 6 = 6$ .
Subtract: $20 - 6 = 14$ .

Answer: $14$

Common mistakes

Doing all multiplication before any division, which breaks the left-to-right rule.
Treating $-3^2$ as $9$ .
Ignoring a fraction bar as a grouping symbol, for example reading $\frac{6 + 4}{2}$ as $6 + 2$ .

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.Evaluate $3 + 6 \times 2^2$ .
Show answer
$27$
2.Evaluate $48 \div 4 \times 3 - 5$ .
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$31$
3.Evaluate $\frac{18 - 3^2}{3} + 1$ .
Show answer
$4$

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