Order of Operations
Order of Operations establishes the standard sequence for simplifying any mathematical expression: Parentheses and grouping symbols first, then Exponents, then Multiplication and Division (left to right), then Addition and Subtraction (left to right) — commonly remembered as PEMDAS. From OpenStax Prealgebra 2E: 10 − 2 · 3 = 4 (not 24) because multiplication precedes subtraction; (10 − 2) · 3 = 24 because parentheses override the default order. This protocol ensures every student and calculator reaches the same answer for a given expression.
Key Concepts
Property When simplifying mathematical expressions perform the operations in the following order: 1. Parentheses and other Grouping Symbols : Simplify all expressions inside parentheses or other grouping symbols, working on the innermost parentheses first. 2. Exponents : Simplify all expressions with exponents. 3. Multiplication and Division : Perform all multiplication and division in order from left to right. These operations have equal priority. 4. Addition and Subtraction : Perform all addition and subtraction in order from left to right. These operations have equal priority. A common way to remember this is the phrase 'Please Excuse My Dear Aunt Sally'.
Examples To simplify $10 2 \cdot 3$, you perform multiplication before subtraction: $10 6 = 4$. To simplify $(10 2) \cdot 3$, you perform the operation in parentheses first: $8 \cdot 3 = 24$. To simplify $5 + (4 1)^2 \div 3$, you start with parentheses ($5+3^2 \div 3$), then exponents ($5+9 \div 3$), then division ($5+3$), and finally addition, which gives $8$.
Explanation This is the official rulebook for solving math problems. Following this order (PEMDAS) ensures everyone gets the same correct answer. Always handle groups first, then powers, then multiplication/division, and finally addition/subtraction from left to right.
Common Questions
What is the correct order of operations?
Parentheses/grouping first, then Exponents, then Multiplication and Division left-to-right, then Addition and Subtraction left-to-right (PEMDAS).
What is 10 − 2 · 3?
Multiply first: 2 · 3 = 6. Then subtract: 10 − 6 = 4. Answer is 4, not 24.
What is (10 − 2) · 3?
Parentheses first: 10 − 2 = 8. Then multiply: 8 · 3 = 24.
How do you simplify 5 + (4−1)² ÷ 3?
Parentheses: 4−1=3. Exponent: 3²=9. Division: 9÷3=3. Addition: 5+3=8.
Are multiplication and division equal in priority?
Yes. When multiplication and division appear together, work left to right — do not always multiply before dividing.
Are addition and subtraction equal in priority?
Yes. Like multiplication and division, addition and subtraction are equal in priority. Evaluate them left to right.