Order of Operations

Why mathematics needs one fixed order of operations, and how to apply it: brackets, then exponents, then multiplication and division, then addition and subtraction.

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Imagine everyone worked out the same expression in a different order. The same question would produce different answers, and mathematics would stop being reliable. That is why there is a fixed order of operations — an agreed sequence that everyone follows.

Concept Why an order is needed

Take the expression 3 + 2 \times 4 . There are two plausible readings:

Adding first: (3 + 2) \times 4 = 5 \times 4 = 20 .
Multiplying first: 3 + (2 \times 4) = 3 + 8 = 11 .

Both are arithmetically sound in isolation, yet they disagree. The convention settles it: multiplication is done before addition, so the correct value is 11.

Concept The rule
1. Brackets 2. Exponents 3. × and ÷ 4. + and −
Work down the list. Everything at one level is finished before the next level begins. The same rule is taught as PEMDAS or BODMAS; the names differ, the order does not.
Step 1 Brackets

Brackets override everything else. Whatever sits inside them is settled first, and only then does the result rejoin the rest of the expression.

(4 + 2) \times 3 = 6 \times 3 = 18
30 - 4 \times (2 + 3) = 30 - 4 \times 5 = 30 - 20 = 10

This is also how you force a different order deliberately: if addition really should come first, put brackets around it.

Step 2 Exponents

Once the brackets are cleared, evaluate any powers.

5 + 2^3 = 5 + 8 = 13
100 \div (2 + 3)^2 = 100 \div 5^2 = 100 \div 25 = 4

Note the second line: the bracket was resolved first, and only then was the power applied to the result.

Step 3 Multiplication and division

These two share a level. Neither outranks the other, so they are carried out left to right in the order they appear.

12 \div 2 \times 3 = 6 \times 3 = 18
Doing the multiplication first would give 12 \div 6 = 2 — a different, incorrect answer.
Step 4 Addition and subtraction

The last level works the same way — equal rank, so left to right.

8 - 3 + 2 = 5 + 2 = 7
Doing the addition first would give 8 - 5 = 3 , which is wrong.

The letters in PEMDAS list M before D and A before S, but that is only for pronunciation. Within a level, position decides.

Example A full evaluation

Evaluate 20 \div 4 + 7 \times 2 - 3 .

No brackets and no exponents, so start at level three.
20 \div 4 = 5 and 7 \times 2 = 14 , giving 5 + 14 - 3 .
Now level four, left to right: 5 + 14 = 19 , then 19 - 3 = 16 .
⟹ 16
Example All four levels at once

Evaluate 2 \times (3 + 4)^2 - 10 .

Brackets: 3 + 4 = 7 , leaving 2 \times 7^2 - 10 .
Exponent: 7^2 = 49 , leaving 2 \times 49 - 10 .
Multiplication: 2 \times 49 = 98 , leaving 98 - 10 .
Subtraction: 98 - 10 = 88 .
⟹ 88

Each line clears exactly one level. Working this way makes an error easy to locate, because every step is small enough to check on its own.

Note Mistakes to avoid
Working strictly left to right and ignoring the levels — 3 + 2 \times 4 is 11, not 20.
Treating multiplication as automatically ahead of division; they share a level.
Treating addition as automatically ahead of subtraction; they share a level too.
Applying an exponent before the bracket beneath it has been resolved.
Clearing several levels in one line, which hides where a slip happened.
Summary
  1. A fixed order exists so that one expression has one answer.
  2. The order is brackets, then exponents, then × and ÷, then + and −.
  3. Multiplication and division share a level and run left to right.
  4. Addition and subtraction share a level and also run left to right.
  5. Brackets are the tool for forcing a different order on purpose.
231. Arithmetic Sequences and Series
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