Order of Operations (PEMDAS)

PEMDAS gives you a strict sequence for evaluating any expression:

1. Parentheses — resolve everything inside grouping symbols first
2. Exponents — handle powers and roots next
3. Multiplication and Division — work through these together, left to right
4. Addition and Subtraction — work through these together, left to right

Two things often surprise students when they first learn this. First, Multiplication and Division share the same priority level — neither beats the other. The same is true for Addition and Subtraction. Second, the order within each pair is determined by position in the expression, not by the letter order in PEMDAS. M appears before D in the acronym only because two letters had to be written in some order — it does not mean multiply before divide.

A reliable way to remember this: PEMDAS describes four tiers, not six separate steps. Tier 1 is parentheses, tier 2 is exponents, tier 3 is multiplication/division (left to right), tier 4 is addition/subtraction (left to right).

Parentheses are the most powerful tool in an expression. They let you override every default priority rule: whatever sits inside them is fully resolved before any operation outside can use the result. This means you can completely change the value of an expression just by adding or moving a pair of parentheses — compare $(3 + 5) \times 2 = 16$ with $3 + 5 \times 2 = 13$.

When parentheses are nested — one set sitting inside another — you always start with the innermost pair and work outward, one layer at a time. Think of it like peeling an onion: the center must be resolved before the next layer makes sense.

For example, in $2 \times (3 + (4 - 1))$: first resolve the inner parentheses, $4 - 1 = 3$. The expression becomes $2\times (3+$. Now resolve the outer parentheses, . Finally, . Trying to handle both layers at once — or starting from the outside — leads to errors because the outer expression depends on the inner result.

After parentheses and exponents, you handle all multiplication and division. The key insight here is that these two operations have equal priority — neither outranks the other. When both appear in an expression, the only rule is to work left to right, performing each operation as you encounter it.

This is where many students go wrong. Because M appears before D in PEMDAS, it's tempting to think multiplication always wins. It doesn't. Consider $12 \div 3 \times 2$. If you multiply first: $12 \div 6 = 2$ — wrong. If you work left to right: $12 \div 3 = 4$, then $4 \times 2 = 8$ — correct.

A useful mental model: once you reach the multiplication/division tier, stop thinking about PEMDAS and simply scan left to right. Every time you see a $\times$ or $\div$, perform it immediately, update the expression, and continue scanning. Don't look ahead.

Addition and subtraction are the last tier. By the time you reach this step, all parentheses have been resolved, all exponents computed, and all multiplication and division handled. What remains is a sequence of additions and subtractions between plain numbers.

Just like multiplication and division, these two operations share equal priority and are handled strictly left to right. The mistake students most often make here is saving all subtractions for last — treating subtraction as something you do after all the additions are complete. This changes the result.

Consider $10 - 4 + 2$. If you add first: $10 - (4 + 2) = 10 - 6 = 4$ — wrong. If you work left to right: $10 - 4 = 6$, then $6 + 2 = 8$ — correct. The subtraction at the beginning of the sequence must be handled before the addition that follows it, because it appears to the left.

Most PEMDAS errors come from three habits worth naming explicitly:

Treating PEMDAS as six separate steps instead of four tiers. Multiplication doesn't always come before division, and addition doesn't always come before subtraction. Once you're inside tier 3 or tier 4, left-to-right position is what determines order — not the letter in the acronym.

Skipping intermediate steps. Multi-operation expressions feel manageable until they're not. Writing each step separately gives you a visible record of where you are. If your answer is wrong, you can trace exactly which step went off track — that's impossible if you've done the whole thing in your head.

Mishandling nested parentheses. Students often resolve all parentheses at the same time, treating inner and outer as equivalent. Remember: the inner set must be fully simplified before the outer set can be touched.

When in doubt, add more parentheses to the original expression to make your intended order explicit. Extra parentheses never change the value of a correctly written expression — they only make the structure clearer.