Order of operations is a set of rules that tells you the correct sequence for evaluating mathematical expressions.

Without rules: 5 + 3 × 2 = ?

Some might think: 5 + 3 = 8, then 8 × 2 = 16 Others might think: 3 × 2 = 6, then 5 + 6 = 11

Which is right? We need a standard order!

PEMDAS: The Order

PEMDAS is an acronym to remember the order:

P - Parentheses (and other grouping symbols) E - Exponents (powers and roots) M - Multiplication D - Division A - Addition S - Subtraction

Memory trick: "Please Excuse My Dear Aunt Sally"

Important: Multiplication/Division Are Equal

Key point: M and D have the SAME priority! Do them left to right as they appear.

Same with A and S - equal priority, left to right.

Better to think of PEMDAS as:

1. Parentheses
2. Exponents
3. MD (Multiplication/Division, left to right)
4. AS (Addition/Subtraction, left to right)

Step 1: Parentheses First

Do operations inside parentheses FIRST.

Example: 3 + (4 × 2)

Step 1: Parentheses → 4 × 2 = 8 Step 2: Addition → 3 + 8 = 11

Answer: 11

Example 2: (5 + 3) × 2

Step 1: Parentheses → 5 + 3 = 8 Step 2: Multiplication → 8 × 2 = 16

Answer: 16

Without parentheses: 5 + 3 × 2 = 5 + 6 = 11 (different!)

Step 2: Exponents

After parentheses, do exponents.

Example: 2 + 3²

Step 1: Exponents → 3² = 9 Step 2: Addition → 2 + 9 = 11

Answer: 11

Example 2: 5 × 2³ - 10

Step 1: Exponents → 2³ = 8 Step 2: Multiplication → 5 × 8 = 40 Step 3: Subtraction → 40 - 10 = 30

Answer: 30

Step 3: Multiplication and Division (Left to Right)

After exponents, do multiplication and division AS THEY APPEAR from left to right.

Example: 12 ÷ 3 × 2

Left to right: 12 ÷ 3 = 4 4 × 2 = 8

Answer: 8

NOT 12 ÷ (3 × 2) = 12 ÷ 6 = 2 ❌

Example 2: 20 × 2 ÷ 5

Left to right: 20 × 2 = 40 40 ÷ 5 = 8

Answer: 8

Step 4: Addition and Subtraction (Left to Right)

Last, do addition and subtraction AS THEY APPEAR from left to right.

Example: 10 - 3 + 2

Left to right: 10 - 3 = 7 7 + 2 = 9

Answer: 9

NOT 10 - (3 + 2) = 10 - 5 = 5 ❌

Example 2: 5 + 8 - 4 + 2

Left to right: 5 + 8 = 13 13 - 4 = 9 9 + 2 = 11

Answer: 11

Complete Example

Evaluate: 3 + 4 × 2² - 5

Step 1: Parentheses (none)

Step 2: Exponents 2² = 4 Expression: 3 + 4 × 4 - 5

Step 3: Multiplication/Division 4 × 4 = 16 Expression: 3 + 16 - 5

Step 4: Addition/Subtraction (left to right) 3 + 16 = 19 19 - 5 = 14

Answer: 14

Nested Parentheses

Work from inside out!

Example: 2 × (3 + (4 × 5))

Step 1: Innermost parentheses 4 × 5 = 20 Expression: 2 × (3 + 20)

Step 2: Next parentheses 3 + 20 = 23 Expression: 2 × 23

Step 3: Multiply 2 × 23 = 46

Answer: 46

Brackets and Braces

Different grouping symbols, same priority as parentheses:

( ) - Parentheses [ ] - Brackets { } - Braces

All mean "do this first!"

Example: 5 + [3 × (2 + 1)]

Inside to out: 2 + 1 = 3 3 × 3 = 9 5 + 9 = 14

Answer: 14

Fraction Bars Act Like Parentheses

The fraction bar groups the numerator and denominator.

Example: (6 + 4) / (5 - 3)

Think of it as: (6 + 4) ÷ (5 - 3)

Numerator: 6 + 4 = 10 Denominator: 5 - 3 = 2 Division: 10 ÷ 2 = 5

Answer: 5

Common Expression: 5 + 3 × 2

Many students incorrectly do this: 5 + 3 = 8 8 × 2 = 16 ❌ WRONG!

Correct order: Multiplication first: 3 × 2 = 6 Then addition: 5 + 6 = 11 ✓

Remember: Multiplication before addition!

Expression with All Operations

Evaluate: 20 ÷ 4 + 3² × 2 - 1

Step 1: Exponents 3² = 9 Expression: 20 ÷ 4 + 9 × 2 - 1

Step 2: Multiplication/Division (left to right) 20 ÷ 4 = 5 9 × 2 = 18 Expression: 5 + 18 - 1

Step 3: Addition/Subtraction (left to right) 5 + 18 = 23 23 - 1 = 22

Answer: 22

Why Left to Right Matters

Example: 8 ÷ 4 × 2

Correct (left to right): 8 ÷ 4 = 2 2 × 2 = 4 ✓

Incorrect (if you did multiplication first): 4 × 2 = 8 8 ÷ 8 = 1 ❌ WRONG!

Always left to right for operations of equal priority!

Negative Numbers in Order of Operations

Example: -3² vs (-3)²

-3²: Exponent first, then negative 3² = 9 Then apply negative: -9

(-3)²: Parentheses include the negative (-3) × (-3) = 9

Very different answers!

Example 2: 5 + -3

This is really 5 + (-3) = 2

Multiple Parentheses

Example: (8 - 3) × (2 + 4)

Step 1: Do both parentheses 8 - 3 = 5 2 + 4 = 6

Step 2: Multiply results 5 × 6 = 30

Answer: 30

Order of Operations with Variables

Example: 2x + 3² when x = 4

Step 1: Substitute 2(4) + 3²

Step 2: Exponents 2(4) + 9

Step 3: Multiplication 8 + 9

Step 4: Addition 17

Answer: 17

Real-World Application

Shopping: You buy 3 shirts at $15 each and a $20 hat. Total cost?

Expression: 3 × 15 + 20

Correct: Multiply first: 3 × 15 = 45 Add: 45 + 20 = 65

Total: $65

If you added first (wrong): 15 + 20 = 35 3 × 35 = 105 ❌ WRONG!

Area and Perimeter

Rectangle: length 5, width (3 + 2)

Perimeter: 2(5 + 3 + 2) Parentheses: 5 + 3 + 2 = 10 Multiply: 2 × 10 = 20

Or: 2 × 5 + 2(3 + 2) = 10 + 2(5) = 10 + 10 = 20

Same answer both ways!

Expressions with Square Roots

Square root acts like parentheses for what's underneath.

Example: √(16 + 9)

Add first: 16 + 9 = 25 Then square root: √25 = 5

Answer: 5

NOT √16 + √9 = 4 + 3 = 7 ❌

Complex Expression Practice

Evaluate: 100 - 4(2³ - 5) + 10 ÷ 2

Step 1: Parentheses (inside first) Exponent in parentheses: 2³ = 8 Subtraction: 8 - 5 = 3 Expression: 100 - 4(3) + 10 ÷ 2

Step 2: Multiplication/Division (left to right) 4(3) = 12 10 ÷ 2 = 5 Expression: 100 - 12 + 5

Step 3: Addition/Subtraction (left to right) 100 - 12 = 88 88 + 5 = 93

Answer: 93

Common Mistakes to Avoid

❌ Mistake 1: Adding before multiplying

• Wrong: 5 + 3 × 2 = 8 × 2 = 16
• Right: 5 + 3 × 2 = 5 + 6 = 11

❌ Mistake 2: Not going left to right

• Wrong: 12 ÷ 3 × 2 = 12 ÷ 6 = 2
• Right: 12 ÷ 3 × 2 = 4 × 2 = 8

❌ Mistake 3: Forgetting parentheses change everything

• 5 + 3 × 2 = 11
• (5 + 3) × 2 = 16 (different!)

❌ Mistake 4: Squaring negative incorrectly

• -3² = -9 (exponent first)
• (-3)² = 9 (parentheses first)

❌ Mistake 5: Not working inside out with nested parentheses

• Must do innermost first!

Memory Aids

PEMDAS: Please Excuse My Dear Aunt Sally

Alternative: GEMDAS

• Grouping symbols (parentheses, brackets, braces)
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

Alternative: BODMAS (British)

• Brackets
• Orders (exponents)
• Division
• Multiplication
• Addition
• Subtraction

All mean the same thing!

Step-by-Step Strategy

To evaluate any expression:

1. Scan for parentheses (or brackets/braces)

   • Do these first, inside to out

2. Look for exponents (and roots)

   • Do these second

3. Find multiplication and division

   • Do left to right

4. Do addition and subtraction

   • Do left to right

5. Check your work

   • Re-read the original problem
   • Make sure you didn't skip anything

Calculator vs Mental Math

Most calculators follow order of operations automatically.

Try: Enter 5 + 3 × 2

Scientific calculator: 11 ✓ Basic calculator might give: 16 ❌ (depends on calculator)

When in doubt, use parentheses to be clear!

Writing Expressions Clearly

Ambiguous: 6 ÷ 2 × 3

Could be interpreted differently. Better to write:

(6 ÷ 2) × 3 = 9 or 6 ÷ (2 × 3) = 1

Use parentheses to make your intent clear!

Practice Pattern Recognition

All × before +:

• 2 + 3 × 4 = 2 + 12 = 14
• 5 × 2 + 1 = 10 + 1 = 11
• 7 + 2 × 5 = 7 + 10 = 17

Parentheses change everything:

• (2 + 3) × 4 = 5 × 4 = 20
• 5 × (2 + 1) = 5 × 3 = 15
• (7 + 2) × 5 = 9 × 5 = 45

Exponents before multiplication:

• 2 × 3² = 2 × 9 = 18
• 5² × 2 = 25 × 2 = 50
• 3 × 2³ = 3 × 8 = 24

Quick Reference

Order:

1. Parentheses (grouping symbols)
2. Exponents (powers, roots)
3. Multiplication/Division (left to right)
4. Addition/Subtraction (left to right)

Remember:

• M and D are equal priority → left to right
• A and S are equal priority → left to right
• Parentheses always first
• Work inside out with nested grouping

PEMDAS = Please Excuse My Dear Aunt Sally

When to Use Parentheses in Writing

To change the order:

• Want addition first? Use parentheses: (5 + 3) × 2

To make clear what you mean:

• Numerator with addition: (2 + 3) ÷ 5
• Multiple operations: (8 - 2) × (3 + 1)

In algebra:

• Distributing: 2(x + 3)
• Negative distributed: -(x - 5)

Summary

Order of operations (PEMDAS) ensures everyone gets the same answer:

Priority from highest to lowest:

1. Parentheses and grouping symbols (inside out)
2. Exponents and roots
3. Multiplication and Division (left to right, equal priority)
4. Addition and Subtraction (left to right, equal priority)

Key principles:

• Parentheses can change the answer completely
• Multiplication/Division before Addition/Subtraction
• Operations of equal priority go left to right
• Fraction bars and square root symbols act as grouping

Applications:

• Evaluating expressions correctly
• Solving equations
• Real-world calculations (shopping, area, etc.)
• Programming and calculators

Common errors to avoid:

• Adding before multiplying
• Not going left to right for M/D and A/S
• Forgetting parentheses change order
• Squaring negatives incorrectly

Master PEMDAS and you'll evaluate any expression correctly every time!

Step 2: Add $5 + 12 = 17$

Answer: $17$

Step 2: Addition $10 + 3 = 13$

Answer: $13$

Step 2: Addition $10 + 3 = 13$

Answer: $13$

Step 2: Brackets $5 - 4 = 1$ $6 + 2[1]$

Step 3: Multiply $2 \times 1 = 2$ $6 + 2$

Step 4: Add $6 + 2 = 8$

Answer: $8$

Step 2: Brackets $5 - 4 = 1$ $6 + 2[1]$

Step 3: Multiply $2 \times 1 = 2$ $6 + 2$

Step 4: Add $6 + 2 = 8$

Answer: $8$