The order of operations is a set of rules for simplifying math expressions in a consistent way. It matters because the same numbers and symbols can give different answers if you do the steps in a different order. PEMDAS helps you remember the priority: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.
Using this order keeps everyone solving the same expression the same way.
PEMDAS is not just a list where every letter always happens separately. Multiplication and division have the same priority, so they are solved from left to right, and addition and subtraction also have the same priority, so they are solved from left to right. Parentheses and other grouping symbols tell you what to simplify first, while exponents act on their base before multiplication or addition.
These rules are used in arithmetic, algebra, formulas, spreadsheets, calculators, and computer programming.
Key Facts
- PEMDAS means Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.
- Grouping symbols are solved first: ( ), [ ], { }, and fraction bars can all group parts of an expression.
- Exponents are evaluated before multiplication, division, addition, and subtraction: 3^2 = 9.
- Multiplication and division have equal priority and are solved from left to right: 24 ÷ 3 × 2 = 8 × 2 = 16.
- Addition and subtraction have equal priority and are solved from left to right: 10 - 4 + 1 = 6 + 1 = 7.
- Example: 6 + 2 × 5^2 = 6 + 2 × 25 = 6 + 50 = 56.
Vocabulary
- Expression
- An expression is a mathematical phrase made of numbers, variables, and operation symbols.
- Operation
- An operation is a math action such as addition, subtraction, multiplication, division, or raising to a power.
- 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 4^3 = 4 × 4 × 4.
- Left-to-right rule
- The left-to-right rule means operations with the same priority are performed in the order they appear from left to right.
Common Mistakes to Avoid
- Doing multiplication before division every time is wrong because multiplication and division have equal priority. Solve them from left to right as they appear.
- Doing addition before subtraction every time is wrong because addition and subtraction have equal priority. Solve them from left to right as they appear.
- Ignoring parentheses is wrong because grouping symbols override the usual order. Always simplify inside parentheses or other grouping symbols first.
- Applying an exponent to the wrong base is wrong because an exponent usually applies only to the number, variable, or grouped expression directly before it. For example, -3^2 = -9, but (-3)^2 = 9.
Practice Questions
- 1 Simplify: 18 - 6 ÷ 3 × 2 + 4.
- 2 Simplify: 5 + 3(8 - 2)^2 ÷ 6.
- 3 Explain why 20 - 4 + 2 is not equal to 20 - (4 + 2), and state the correct value of each expression.