Order of Operations Explained- Preparing for Your Math Test Success
What the Order of Operations Actually Is
Your math teacher throws this at you: 6 + 2 × 5 = ?
Wrong answer? 40. Right answer? 16.
The order of operations is the set of rules telling you which calculation to do first. Without it, every problem becomes a mess of competing interpretations. With it, math problems have exactly one correct answer.
This isn't optional knowledge. It shows up on every standardized test, every algebra class, and every time you need math to actually work. Learn it once, stop making the same dumb mistakes forever.
The Acronym That Saves You
Most people learn PEMDAS:
- Parentheses
- Exponents (powers and roots)
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)
Some countries use BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction). Same rules, different name.
Here's what most students miss: multiplication doesn't always come before division, and addition doesn't always come before subtraction. You work left to right when you hit those pairs.
Breaking Down Each Step
Parentheses Come First, Always
Calculate everything inside parentheses before touching anything else. Nested parentheses? Start from the innermost and work your way out.
Example: 2 × (3 + 4) = 2 × 7 = 14 ✅
Not 2 × 3 + 4 = 6 + 4 = 10 ❌
Exponents Are Next
Handle powers and roots after parentheses but before anything else.
Example: 3² + 6 = 9 + 6 = 15 ✅
Not 3² + 6 = 3 × 8 = 24 ❌
Multiplication and Division: Left Wins
When you see both in the same problem, calculate from left to right. The same rule applies to addition and subtraction.
Example: 8 ÷ 4 × 2 = 2 × 2 = 4 ✅
Not 8 ÷ (4 × 2) = 8 ÷ 8 = 1 ❌
You only group multiplication together if parentheses force you to.
Addition and Subtraction: Same Deal
Work left to right through the whole chain.
Example: 10 - 3 + 2 = 7 + 2 = 9 ✅
Not 10 - (3 + 2) = 10 - 5 = 5 ❌
How to Actually Solve These Problems
Here's the step-by-step process for any expression:
- Scan for parentheses. Solve inside them first.
- Find all exponents. Calculate powers and roots.
- Work through multiplication and division. Left to right.
- Finish with addition and subtraction. Left to right.
- Check your work. Does the answer make sense?
Let's walk through a real problem:
Problem: 3 + (2 × 4²) - 6 ÷ 3
Step 1: Parentheses → 3 + (2 × 16) - 6 ÷ 3
Step 2: Exponents inside parentheses → 3 + (2 × 16) - 6 ÷ 3
Step 3: Multiplication inside parentheses → 3 + 32 - 6 ÷ 3
Step 4: Division → 3 + 32 - 2
Step 5: Addition and subtraction left to right → 35 - 2 = 33
Where Students Screw Up
These mistakes show up constantly on tests:
- Doing operations in the wrong order because it "feels faster"
- Assuming multiplication always beats division — it doesn't
- Ignoring nested parentheses — solve inside-out
- Skipping steps on easy problems — that's exactly when you make errors
The fix? Write every single step. No shortcuts until you've done it the long way 50 times.
Quick Reference Table
| Symbol | Name | When to Calculate |
|---|---|---|
| ( ) | Parentheses / Brackets | First — always |
| x², √ | Exponents / Orders | Second |
| ×, ÷ | Multiplication, Division | Third — left to right |
| +, − | Addition, Subtraction | Last — left to right |
Practice Problems
Try these before checking answers:
1. 5 + 3 × 4 = ?
2. (5 + 3) × 4 = ?
3. 20 ÷ 5 × 2 = ?
4. 2³ + 10 ÷ 2 = ?
5. 7 - 3 × 2 + 8 ÷ 4 = ?
Answers: 1) 17 | 2) 32 | 3) 8 | 4) 12 | 5) 4
The Bottom Line
Order of operations isn't complicated. There are four steps. Parentheses first, then exponents, then multiplication/division left to right, then addition/subtraction left to right.
The only thing between you and correct answers is discipline. Write your steps. Don't skip ahead. Check your work.
That's it. Now go practice.