Order of Operations (PEMDAS / BODMAS): Why the Order Matters

What PEMDAS and BODMAS actually mean

PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. BODMAS stands for Brackets, Orders, Division, Multiplication, Addition, Subtraction. The words differ but the rule is identical: brackets are the same as parentheses, and orders are the same as exponents. Both are just memory aids for one agreed order.

One point trips people up: multiplication and division sit at the same level, and so do addition and subtraction. Within a level you work strictly left to right. So 8 ÷ 2 × 4 is not 1 — you do 8 ÷ 2 = 4 first, then 4 × 4 = 16. The letter order in PEMDAS does not mean multiplication always beats division.

Why a shared order is necessary

Take 3 + 4 × 2. If you read left to right and add first, you get 7 × 2 = 14. If you multiply first, you get 3 + 8 = 11. Both look reasonable, yet only one can be the accepted answer. The order of operations settles the tie: multiplication comes before addition, so the correct value is 11.

This is not just a school convention — calculators, spreadsheets, and programming languages all follow it. Without an agreed order, the same formula written once would compute differently on every machine. The rule is what lets a formula mean exactly one thing, whoever reads it.

A full worked example

Evaluate 6 + 2 × (3 + 1)² ÷ 4. Start inside the parentheses: 3 + 1 = 4, giving 6 + 2 × 4² ÷ 4. Next the exponent: 4² = 16, giving 6 + 2 × 16 ÷ 4. Now multiplication and division left to right: 2 × 16 = 32, then 32 ÷ 4 = 8, giving 6 + 8. Finally the addition: 6 + 8 = 14.

Notice how each pass removes one level of the hierarchy: parentheses, then exponent, then the multiply-divide pair, then the add-subtract pair. Working one level at a time and left to right within it is the whole method — do that and every expression collapses to a single number.

Frequently asked questions

Are PEMDAS and BODMAS different rules?

No. They are the same rule with different acronyms. Brackets equal parentheses and orders equal exponents, so both give the identical evaluation order.

Does multiplication always come before division?

No. They share one level and are done left to right. In 8 ÷ 2 × 4 the division comes first because it appears first, giving 16.

What about nested parentheses?

Work from the innermost pair outward. Evaluate the deepest bracket first, replace it with its value, then handle the bracket around it, and so on.

Where does a negative sign or a root fit in?

A root is an exponent (a square root is the ½ power), so it sits at the exponent level. A leading negative sign acts like multiplying by −1 and is applied at the multiplication level.