Order of Operations- PEMDAS Rules Explained
What Is PEMDAS and Why It Matters
PEMDAS is an acronym that tells you the correct order to solve math problems. Without it, the same equation gives different answers depending on who solves it.
Math teachers use PEMDAS so everyone gets the same result. Engineers use it to build safe structures. Programmers use it to write working code. You need it for every math class from middle school onward.
The Six Letters Explained
Each letter stands for a step. You work through them from left to right:
- Parentheses — solve anything inside ( ) first
- Exponents — powers and roots come next
- Multiplication and Division — do these in order from left to right
- Addition and Subtraction — finish with these, left to right
The tricky part most people miss: M and D have equal priority. So do A and S. You don't finish all multiplication before division. You work left to right, doing whichever operation appears first.
Left to Right Rule: The Part Everyone Gets Wrong
Students remember "multiply before divide" and "add before subtract." That's wrong. Here's the reality:
For 8 ÷ 4 × 2, you don't do multiplication first. You go left to right. 8 ÷ 4 = 2, then 2 × 2 = 4. Wrong approaches give you 1.
Same deal with 10 - 3 + 2. You don't subtract first because subtraction comes last in PEMDAS. You work left to right: 10 - 3 = 7, then 7 + 2 = 9.
Step-by-Step Examples
Example 1: Basic PEMDAS
Solve: 3 + 6 × (5 + 2) ÷ 3 - 1
Step 1 — Parentheses: 3 + 6 × 7 ÷ 3 - 1
Step 2 — Exponents: None here, skip it
Step 3 — Multiplication and Division left to right: 6 × 7 = 42, then 42 ÷ 3 = 14
Now you have: 3 + 14 - 1
Step 4 — Addition and Subtraction left to right: 3 + 14 = 17, then 17 - 1 = 16
Answer: 16
Example 2: Nested Parentheses
Solve: 2 × (3 + (4 - 1))
Work inside the innermost parentheses first: 4 - 1 = 3
Now you have: 2 × (3 + 3)
Then the outer parentheses: 3 + 3 = 6
Finish: 2 × 6 = 12
Answer: 12
Example 3: Exponents Included
Solve: 2³ + 4 × (6 - 2)²
Parentheses first: 6 - 2 = 4
Exponents next: 2³ = 8 and 4² = 16
Now: 8 + 4 × 16
Multiplication: 4 × 16 = 64
Addition: 8 + 64 = 72
Answer: 72
PEMDAS vs. GEMDAS
Some textbooks use GEMDAS instead. The G stands for Grouping, which is the same idea as Parentheses. Everything else works identically.
Other variations include BOMDAS (Brackets, Orders, Multiplication, Division, Addition, Subtraction) used in some countries. The concept stays the same—only the names change.
Common Mistakes to Avoid
- Doing all multiplication before division — work left to right
- Doing all addition before subtraction — work left to right
- Ignoring nested parentheses — always solve innermost first
- Skipping the order entirely — solving left to right like reading a book
- Forgetting exponents exist — they come second, after parentheses
PEMDAS vs. Standard Left-to-Right Solving
| Expression | Correct Answer | Wrong Answer (Left-to-Right) |
|---|---|---|
| 8 ÷ 4 × 2 | 4 | 1 |
| 10 - 3 + 2 | 9 | 5 |
| 6 + 2 × 5 | 16 | 40 |
| 20 - 4 × 3 + 1 | 9 | 49 |
The left-to-right approach looks natural when you read it. That's exactly why it produces wrong answers.
How to Get Started
Follow this checklist every time you face a complex expression:
- Scan the entire problem first — identify all parentheses and exponents
- Circle or highlight parentheses groups — solve each one separately
- Look for exponents — solve all powers before moving on
- Go left to right through multiplication and division — don't batch them
- Go left to right through addition and subtraction — don't batch these either
- Write out each step — don't try to do multiple steps at once
Practice with simple expressions first. Once you can solve 3 + 4 × 2 without hesitation, move to problems with two or three operations. Build up to nested parentheses and exponents.
When PEMDAS Gets Complicated
Some expressions use brackets [ ] or braces { } alongside parentheses. All grouping symbols work the same way—solve the innermost first, then work outward.
Absolute value bars | | also count as grouping symbols. Fraction bars act as implicit parentheses around the numerator and denominator.
These variations don't change PEMDAS. They just give you more grouping symbols to handle before moving to exponents.
The Bottom Line
PEMDAS isn't optional. It's the agreed-upon system that makes math consistent. Parentheses first, then exponents, then multiplication and division left to right, then addition and subtraction left to right.
Write out every step until it becomes automatic. The goal is solving problems correctly, not appearing fast. Speed comes after you stop making errors.