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Foundations · Arithmetic

Order of operations

Brackets, powers, multiply and divide, add and subtract — the one rule that keeps a long expression from giving three different answers.

First, the rules

One order, so everyone gets the same answer

2 + 3 × 4 is 14 everywhere in the world, because multiplication is done before addition. The order is a convention — but a convention every calculator and every exam follows.

★ The order
Examples 1–3
1. Brackets 2. pOwers (indices, roots) 3. Divide and Multiply, left to right 4. Add and Subtract, left to right
BODMAS, BIDMAS, PEMDAS — same rule, different letters.
★ Same level → left to right
Example 2
12 ÷ 3 × 2 = 4 × 2 = 8 (not 12 ÷ 6 = 2)
10 − 4 − 3 = 6 − 3 = 3 (not 10 − 1 = 9)
★ Hidden brackets
Example 3
A fraction bar brackets the top and the bottom: 8 + 42 + 1 = 123 = 4. A root brackets what is under it: √(9 + 16) = 5, not 3 + 4.
Nested brackets
Example 3
Work from the inside out: 2 × [3 + (8 − 5)²] = 2 × [3 + 9] = 24.
Negative numbers and powers
−3² = −9 but (−3)² = 9 — the power binds before the minus sign unless the bracket says otherwise.

See it

Watch an expression collapse

Pick a shape, set the numbers, and press "Next step": one operation is done at a time, always the highest-priority one left.

Step by step

Worked examples

1

Three expressions, three answers

order

Evaluate (a) 5 + 3 × 2, (b) (5 + 3) × 2, (c) 5 + 3².

Ask yourself: Which operation has the highest priority in each one?

Show the solution
  1. (a) Multiply first: 3 × 2 = 6, then 5 + 6 = 11.
  2. (b) Brackets first: 5 + 3 = 8, then 8 × 2 = 16.
  3. (c) Power first: 3² = 9, then 5 + 9 = 14.

Answer 11 · 16 · 14

2

Left to right matters

same level

Evaluate 36 ÷ 6 × 3 − 4 + 2.

Ask yourself: Division and multiplication are a tie — so is subtraction and addition. Which comes first in a tie?

Show the solution
  1. Left to right among ÷ and ×: 36 ÷ 6 = 6, then 6 × 3 = 18.
  2. Then left to right among − and +: 18 − 4 = 14, then 14 + 2 = 16.
  3. Doing 6 × 3 first (36 ÷ 18 = 2) or 4 + 2 first (18 − 6 = 12) both give wrong answers.

Answer 16

3

Hidden brackets

fraction bar · root · nesting

Evaluate 2 × (7 − 3)²4 + 4 + √25 − 9.

Ask yourself: Where are the invisible brackets?

Show the solution
  1. Top of the fraction: bracket 7 − 3 = 4, power 4² = 16, multiply 2 × 16 = 32. Bottom: 4 + 4 = 8. Fraction 32 ÷ 8 = 4.
  2. Under the root: 25 − 9 = 16, and √16 = 4.
  3. Add: 4 + 4 = 8.

Answer 8

Practice: evaluate it

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Brackets, powers, left-to-right ties — the wrong answers in the options are the classic mistakes. Your best score is kept in this browser only.

About this lesson

Order of operations (BODMAS / PEMDAS): why 2 + 3 × 4 is 14 and not 20, how brackets change everything, where powers fit, why multiplication and division are worked left to right, and the fraction bar as a hidden bracket. A step-by-step evaluator you can control, worked examples and practice questions. Everything on this page runs in your browser; nothing you type is sent or stored. Spotted an error? Tell us.