Order of operations is the set of rules that tells you which parts of a math expression to simplify first. It matters because the same symbols can give different answers if you do the steps in a different order. PEMDAS is a memory aid: Parentheses, Exponents, Multiply, Divide, Add, Subtract.
The goal is not just to memorize a phrase, but to read expressions in a consistent way that everyone agrees on.
Start inside parentheses or other grouping symbols, then simplify exponents, then handle multiplication and division from left to right, and finally handle addition and subtraction from left to right. Multiplication does not always come before division, and addition does not always come before subtraction. Operations at the same level are completed in the order they appear from left to right.
For example, 2 + 3 × 4 becomes 2 + 12 = 14 because multiplication has higher priority than addition.
Understanding Math: Order of operations (PEMDAS)
An expression is more like a sentence than a list of chores. Its parts are connected in layers. Grouping symbols create a smaller expression that acts as one number in the larger expression.
A fraction bar does this too. Everything above the bar belongs together, and everything below it belongs together. This is why a long fraction can be confusing when copied onto one line.
Writing clear groups prevents a reader from attaching a number to the wrong part. When groups are nested, work from the innermost group outward. Each completed group can then be replaced by its value, making the remaining work easier to read.
Exponents deserve special care because they describe repeated multiplication, not ordinary multiplication by the exponent. Five squared means five multiplied by five. It does not mean five times two.
Negative signs can make this stage tricky. A leading negative sign may represent subtraction, or it may be part of a negative number. By convention, negative three squared means the negative of three squared, which gives negative nine.
The quantity negative three in parentheses, squared, gives positive nine. Those two statements look similar when written with symbols, yet their meanings differ. Careful grouping removes the ambiguity.
The left to right rule exists because division and subtraction are not freely reversible. For example, starting with sixty, dividing by three, then multiplying by five gives one hundred. Multiplying by five first would create a different calculation.
Division can be understood as multiplication by a reciprocal, while subtraction can be understood as adding an opposite value. These ideas explain why they share a level with multiplication and addition. They are related operations, but their order still matters.
Do not combine nearby numbers just because they look convenient. Combine them only when the expression structure permits it.
Students meet these rules in formulas, shopping calculations, spreadsheets, computer code, and science work. A physics formula may contain a squared speed inside a larger calculation. A spreadsheet may calculate a discount before adding tax.
A calculator usually follows standard precedence, but its display may hide how it grouped your input. Adding grouping symbols makes intended steps visible and protects against typing mistakes. A useful habit is to copy the expression, simplify one layer per line, and keep the operation signs attached to the numbers they belong with.
Estimate the size and sign of the answer before finishing. An answer that is wildly too large, too small, or has an unexpected negative sign often points to a skipped group or a mistaken exponent.
Key Facts
- PEMDAS stands for Parentheses, Exponents, Multiply, Divide, Add, Subtract.
- Parentheses and other grouping symbols are simplified first: ( ), [ ], { }, and fraction bars.
- Exponents are simplified before multiplication, division, addition, and subtraction: 3^2 + 4 = 9 + 4 = 13.
- Multiplication and division have equal priority and are done left to right: 24 ÷ 3 × 2 = 8 × 2 = 16.
- Addition and subtraction have equal priority and are done left to right: 10 - 4 + 1 = 6 + 1 = 7.
- Example rule: 2 + 3 × 4 = 2 + 12 = 14, not 20.
Vocabulary
- Expression
- An expression is a mathematical phrase with numbers, variables, and operations but no equal sign.
- Operation
- An operation is a mathematical action such as addition, subtraction, multiplication, division, or exponentiation.
- Parentheses
- Parentheses are grouping symbols that tell you to simplify the expression inside them first.
- Exponent
- An exponent tells how many times a base is used as a factor, such as 2^3 = 2 × 2 × 2.
- Left to right
- Left to right means simplifying operations of the same priority in the order they appear across the page.
Common Mistakes to Avoid
- Doing multiplication before division every time is wrong because multiplication and division have equal priority. Work left to right when both appear.
- Doing addition before subtraction every time is wrong because addition and subtraction have equal priority. Work left to right when both appear.
- Adding before multiplying in expressions like 2 + 3 × 4 is wrong because multiplication has higher priority than addition. The correct first step is 3 × 4.
- Ignoring parentheses is wrong because grouping symbols change the order of evaluation. In (2 + 3) × 4, the addition must happen before the multiplication.
Practice Questions
- 1 Evaluate 18 ÷ 3 × 2 + 5.
- 2 Evaluate 4 + 2^3 × (10 - 7).
- 3 Explain why 12 - 4 + 2 is not solved by doing 4 + 2 first, even though addition appears in PEMDAS before subtraction.