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Order of Operations (PEMDAS)

Order of Operations is the set of rules that tells you which calculations to do first when an expression has more than one operation. The acronym PEMDAS helps you remember the order: Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right).

The order of operations is a convention used in mathematics to ensure that any numerical expression is evaluated consistently and yields a single, unambiguous result. Operations are performed in the following hierarchy: (1) expressions inside Parentheses or other grouping symbols, (2) Exponents and roots, (3) Multiplication and Division from left to right, and (4) Addition and Subtraction from left to right. Multiplication does not take priority over division, nor does addition take priority over subtraction — operations at the same level are simply worked from left to right.

Worked Example

Problem: Evaluate: 3+4×(6−2)2÷83 + 4 \times (6 - 2)^2 \div 8
Parentheses: Simplify inside the parentheses first.
3+4×(4)2÷83 + 4 \times (4)^2 \div 8
Exponents: Next, evaluate the exponent.
3+4×16÷83 + 4 \times 16 \div 8
Multiplication and Division (left to right): Multiply 4 × 16, then divide the result by 8. Work left to right.
3+64÷8=3+83 + 64 \div 8 = 3 + 8
Addition: Finally, add.
3+8=113 + 8 = 11
Answer: The value of the expression is 1111.

Why It Matters

Without a standard order of operations, the same expression could produce different answers depending on who reads it. Engineers, programmers, scientists, and accountants all rely on these rules so that formulas and calculations mean the same thing to everyone. Whenever you type a formula into a spreadsheet or calculator, the software follows PEMDAS behind the scenes.

Common Mistakes

Mistake: Thinking multiplication always comes before division (or addition always before subtraction) because of the letter order in PEMDAS.
Correction: Multiplication and division are equal in priority — you handle them left to right. The same goes for addition and subtraction. For example, in 8÷2×48 \div 2 \times 4, you divide first (getting 4) and then multiply (getting 16), not the other way around.
Mistake: Ignoring nested parentheses and working from the outside in.
Correction: Always simplify the innermost grouping symbols first, then work outward. For example, in 2×[3+(5−1)]2 \times [3 + (5 - 1)], compute (5−1)=4(5 - 1) = 4 first, then 3+4=73 + 4 = 7, and finally 2×7=142 \times 7 = 14.

Related Terms

  • Expression — What the order of operations is applied to
  • Evaluate — The process of finding an expression's value
  • Parentheses — Grouping symbols handled first in PEMDAS
  • Brackets — Another grouping symbol treated like parentheses