Elementary Operations Sequence- Step-by-Step Guide

What Is the Elementary Operations Sequence?

The elementary operations sequence is the rule that tells you which math operation to solve first in an expression. Without it, you'd get different answers every time.

It's commonly remembered as PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) or BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction). Same thing, different acronyms.

The Actual Order (Don't Memorize It Wrong)

Here's where people mess up. Multiplication doesn't always come before division. Same with addition and subtraction. The rule is:

• Parentheses/Brackets first — solve anything inside () or []
• Exponents/Orders second — powers, square roots
• Multiplication and Division — work left to right
• Addition and Subtraction — work left to right

The last two groups go by appearance order, not by name priority.

Why This Matters

Get this wrong and your answer is wrong. Full stop. It affects everything from basic homework to complex engineering calculations.

Quick Comparison: PEMDAS vs BODMAS

Common Mistakes That Kill Your Answer

1. Multiplying before dividing

If your expression has "÷ 2 × 3", you do division first because it appears left. Not because division beats multiplication.

2. Adding before subtracting

Same logic. "5 - 3 + 2" equals 4, not 0. You work left to right.

3. Ignoring nested parentheses

Solve inside-out. Inner parentheses first, then outer ones.

4. Skipping exponents

3 + 2³ isn't 5³. It's 3 + 8 = 11. Powers come before addition.

Getting Started: How to Solve Any Expression

Step 1: Scan the whole expression. Find all parentheses and brackets.

Step 2: Solve everything inside them. If there are nested ones, go inner to outer.

Step 3: Look for exponents or roots. Handle those next.

Step 4: Scan left to right for multiplication and division. Solve each as you hit it.

Step 5: Scan left to right for addition and subtraction. Solve each as you hit it.

Step 6: You should have one number left. That's your answer.

Worked Example

Solve: 3 + (4 × 2²) - 6 ÷ 3

Step 1 — Parentheses: 3 + (4 × 2²) - 6 ÷ 3

Step 2 — Inside parentheses, exponents: 2² = 4

Step 3 — Inside parentheses, multiplication: 4 × 4 = 16

Result so far: 3 + 16 - 6 ÷ 3

Step 4 — Division (left to right): 6 ÷ 3 = 2

Result: 3 + 16 - 2

Step 5 — Addition and subtraction (left to right): 3 + 16 = 19, then 19 - 2 = 17

Practice Problem

Try this one: (8 + 2) × 3² ÷ 5 - 4

Answer below.

Answer: (8 + 2) = 10. 3² = 9. 10 × 9 = 90. 90 ÷ 5 = 18. 18 - 4 = 14.

Memorization Trick That Actually Sticks

Forget singing the alphabet. Try this sentence:

"Please Excuse My Dear Aunt Sally"

Each word starts with the right letter in order. That's it. That's the whole mnemonic.

When to Use the Sequence in Real Life

• Building cost estimates with multiple discounts
• Calculating recipe portions with scaling factors
• Programming logic and code evaluation
• Any spreadsheet formula with mixed operations

You use this every day without realizing it. The calculator on your phone uses it. Excel uses it. Your order matters.