Order of operations

The four levels

Work through an expression level by level: first anything inside brackets; then powers and roots; then multiplication and division; finally addition and subtraction.

Operations on the same level are done from left to right. Multiplication does not come before division, and addition does not come before subtraction — they share a level. So 12 \div 3 \times 2 = 4 \times 2 = 8, not 12 \div 6 = 2.

Powers and negative signs

A power applies only to what is immediately beneath it. In -3^2 the exponent belongs to 3 alone, so the value is -(3^2) = -9. To square the negative number write (-3)^2 = 9.

Stacked powers are evaluated from the top down: 2^{3^2} = 2^9 = 512, not 8^2 = 64.

Fraction bars and roots are brackets in disguise

A fraction bar groups its numerator and its denominator, and a root sign groups everything under it. When typing on one line you must add those brackets yourself: \frac{6 + 2}{4} becomes (6 + 2) / 4.

Worked examples

Common mistakes

• Working strictly left to right: 2 + 3 \times 4 is 2 + 12 = 14, not 5 \times 4 = 20.
• Treating “multiplication before division” as a rule. They have equal priority and go left to right.
• Reading -3^2 as $9$. Without brackets the square applies before the minus sign.
• Forgetting the brackets a fraction bar implies when typing a fraction on one line.

Practice problems

1. 1.6 + 4 \times 5Show answer
2. 2.(6 + 4) \times 5Show answer
3. 3.18 \div 3 \times 2 - 4Show answer
4. 4.-2^4 + (−2)^4Show answer