Unit 1: Foundations of Algebra
Unit 1 - Section 2: Order of Operations
Article: Mastering the Order of Operations
In algebra (and all of mathematics), we often encounter expressions that contain multiple operations. To ensure consistency and correctness, we follow a specific sequence known as the Order of Operations. This order dictates which operations to perform first, second, and so on.
Why Is the Order of Operations Important?
Consider the following expression:
8 + 4 × 2
If you add first: (8 + 4) × 2 = 12 × 2 = 24
If you multiply first: 8 + (4 × 2) = 8 + 8 = 16
The two results are different! Without a standard order, everyone might get a different answer. The Order of Operations eliminates this confusion.
The Standard Order: PEMDAS
We follow the PEMDAS rule:
P: Parentheses (or brackets)
E: Exponents (powers and roots)
MD: Multiplication and Division (from left to right)
AS: Addition and Subtraction (from left to right)
Breaking Down PEMDAS
1. Parentheses
Perform calculations inside parentheses or grouping symbols first.
This also includes brackets [ ] and braces { }.
Example:
(3 + 5) × 2 = 8 × 2 = 16
2. Exponents
Apply any powers or roots.
Example:
2³ = 2 × 2 × 2 = 8
3. Multiplication and Division
Work from left to right. These operations have equal priority.
Example:
16 ÷ 4 × 2 = (16 ÷ 4) × 2 = 4 × 2 = 8
4. Addition and Subtraction
Work from left to right. These also have equal priority.
Example:
10 - 3 + 2 = (10 - 3) + 2 = 7 + 2 = 9
Special Notes
When expressions contain nested parentheses, work from the innermost outward.
If multiple operations of the same rank occur (such as multiplication and division), process them from left to right.
Real-Life Applications of Order of Operations
Calculating taxes, discounts, and totals in finance
Computer programming algorithms
Physics formulas
Construction measurements
