Grouping Symbols
Hover over a variable in the formula above, or see glossary below:
Any expression inside parentheses (), brackets [], or braces {} must be evaluated before anything outside of them. If there are nested symbols, evaluate from the inside out.
Exponents & Powers
After parentheses, evaluate any terms with exponents or roots. These represent repeated multiplication, so they carry more weight than standard multiplication.
Left-to-Right Operations
Multiplication and Division share the same priority. Evaluate them from left to right. Then, Addition and Subtraction share the lowest priority, and are also evaluated from left to right.
Arithmetic and Order of Operations
The order of operations is a collection of rules that dictate which procedures to perform first in order to evaluate a given mathematical expression. Without a standard order of operations, a single expression could yield multiple different, yet seemingly valid, results.
The PEMDAS Rule
In the United States, the acronym PEMDAS is commonly used as a mnemonic to remember the order of operations:
- Parentheses: Evaluate expressions inside grouping symbols like
(),[], or{}. - Exponents: Evaluate powers and roots.
- Multiplication and Division: Evaluate these operations from left to right.
- Addition and Subtraction: Evaluate these operations from left to right.
The Left-to-Right Convention
A common misconception is that multiplication always precedes division, or that addition always precedes subtraction. In reality, Multiplication and Division are on the same "level" of priority. When both appear in an expression, you simply evaluate them from left to right. The same applies to Addition and Subtraction.
Consider the expression:
- Incorrect (Multiply first):
- Correct (Left to right):
Nested Grouping Symbols
When an expression contains multiple sets of parentheses, such as , always start with the innermost set and work outwards.
References
- Khan Academy: Order of Operations
- Common Core State Standards for Mathematics: CCSS.5.OA.A.1, CCSS.6.EE.A.2.c
AI Assistance Disclaimer: This module uses AI-assisted educational models and interactive visual representations to help explain scientific and mathematical concepts. For formal research or academic evaluation, please verify formulas and data against standard primary reference materials.