Order of Operations (PEMDAS)

Master the fundamental rules that determine the correct sequence for solving mathematical expressions. Understanding PEMDAS is essential for accurate calculations in algebra and beyond.

What is PEMDAS?

PEMDAS is an acronym that helps you remember the correct order to perform operations in a mathematical expression. Without following this order, you might get completely different (and incorrect) answers!

P

Parentheses

( )

E

Exponents

x²

M

Multiplication

×

D

Division

÷

A

Addition

+

S

Subtraction

−

💡 Memory Aids

Popular phrases to remember PEMDAS:

Please Excuse My Dear Aunt Sally
Purple Elephants March Down Avenues Saturday

🌍 Other Names

Different countries use different acronyms:

BODMAS (UK): Brackets, Orders, Division, Multiplication, Addition, Subtraction
BIDMAS (UK): Brackets, Indices, Division, Multiplication, Addition, Subtraction
BEDMAS (Canada): Brackets, Exponents, Division, Multiplication, Addition, Subtraction

The Rules Explained

Let's break down each step of PEMDAS and understand when and how to apply each operation:

1. Parentheses First

Always solve what's inside parentheses (or brackets) before anything else. If there are nested parentheses, work from the innermost to the outermost.

3 × (4 + 2)

↓

3 × 6

↓

18

2. Exponents Second

After parentheses, calculate any exponents (powers or roots).

2 + 3²

↓

2 + 9

↓

11

3. Multiplication and Division (Left to Right)

Multiplication and division have equal priority. Perform them from left to right as they appear.

20 ÷ 4 × 2

↓

5 × 2

↓

10

⚠️ Critical Point

Many people think multiplication always comes before division because M comes before D in PEMDAS. This is incorrect! They have equal priority and are performed left to right.

4. Addition and Subtraction (Left to Right)

Finally, perform addition and subtraction from left to right. Like multiplication and division, these operations have equal priority.

10 - 3 + 2

↓

7 + 2

↓

9

Worked Examples

Let's work through examples of increasing complexity to solidify your understanding:

Example 1: Simple Expression

5 + 3 × 2

Step 1: Identify operations: Addition and Multiplication

Step 2: Multiplication first: 3 × 2 = 6

Step 3: Then addition: 5 + 6 = 11

⚠️ Common mistake: 5 + 3 = 8, then 8 × 2 = 16 (WRONG!)

Example 2: With Parentheses

(8 + 2) × 3 - 4

Step 1: Parentheses first: (8 + 2) = 10

Step 2: Expression becomes: 10 × 3 - 4

Step 3: Multiplication: 10 × 3 = 30

Step 4: Subtraction: 30 - 4 = 26

Example 3: With Exponents

2 + 3² × 4 - 1

Step 1: Exponents first: 3² = 9

Step 2: Expression becomes: 2 + 9 × 4 - 1

Step 3: Multiplication: 9 × 4 = 36

Step 4: Expression becomes: 2 + 36 - 1

Step 5: Left to right: 2 + 36 = 38

Step 6: Then: 38 - 1 = 37

Example 4: Complex Expression

48 ÷ (2 + 4) × 2³ - 5

Step 1: Parentheses: (2 + 4) = 6

Step 2: Exponents: 2³ = 8

Step 3: Expression becomes: 48 ÷ 6 × 8 - 5

Step 4: Division (left to right): 48 ÷ 6 = 8

Step 5: Multiplication: 8 × 8 = 64

Step 6: Subtraction: 64 - 5 = 59

Common Mistakes to Avoid

Even experienced students make these errors. Learn to recognize and avoid them:

❌ Mistake 1: Ignoring the Order

Wrong: 6 + 4 × 2 = 10 × 2 = 20

Correct: 6 + 4 × 2 = 6 + 8 = 14

Always do multiplication before addition!

❌ Mistake 2: Multiplication Before Division (Always)

Wrong: 12 ÷ 2 × 3 = 12 ÷ 6 = 2

Correct: 12 ÷ 2 × 3 = 6 × 3 = 18

Multiplication and division have equal priority—work left to right!

❌ Mistake 3: Forgetting Exponents

Wrong: 5 + 2³ × 2 = 7³ × 2 = 343 × 2 = 686

Correct: 5 + 2³ × 2 = 5 + 8 × 2 = 5 + 16 = 21

Calculate exponents before multiplication and addition!

❌ Mistake 4: Addition Before Subtraction (Always)

Wrong: 10 - 3 + 2 = 10 - 5 = 5

Correct: 10 - 3 + 2 = 7 + 2 = 9

Addition and subtraction have equal priority—work left to right!

Special Cases

Some situations require extra attention when applying the order of operations:

Nested Parentheses

When parentheses are inside other parentheses, work from the innermost pair outward.

📝 Example: 2 × [3 + (4 × 2)]

Step 1: Innermost parentheses: (4 × 2) = 8

Step 2: Expression becomes: 2 × [3 + 8]

Step 3: Brackets: [3 + 8] = 11

Step 4: Multiplication: 2 × 11 = 22

Implied Multiplication

When a number is written directly next to parentheses, multiplication is implied.

📝 Example: 4(3 + 2)

Note: 4(3 + 2) means 4 × (3 + 2)

Step 1: Parentheses: (3 + 2) = 5

Step 2: Multiplication: 4 × 5 = 20

Fractions as Division

A fraction bar acts like both parentheses and division. Solve the numerator and denominator separately, then divide.

📝 Example: (6 + 4) / (3 - 1)

Step 1: Numerator: 6 + 4 = 10

Step 2: Denominator: 3 - 1 = 2

Step 3: Division: 10 ÷ 2 = 5

💡 Pro Tip

When in doubt, use parentheses to make your intentions clear! Extra parentheses never hurt, and they can prevent confusion.

Practice Exercises

Test your understanding with these practice problems:

Exercise 1: 8 + 2 × 5

Exercise 2: (6 + 3) × 2

Exercise 3: 15 - 3 × 2 + 4

Exercise 4: 4² + 3 × 2

Exercise 5: 20 ÷ 4 × 5

Exercise 6: 3 + 2 × (8 - 3)

Exercise 7: 100 - 10 × 5 + 20

Exercise 8: 2³ + 3² - 1

Exercise 9: 5(2 + 3)

Exercise 10: 18 ÷ 3 + 4 × 2 - 1

🎉 Congratulations!

You've completed the Order of Operations (PEMDAS) tutorial. You now know:

✓ What PEMDAS stands for and why it matters
✓ The correct order to perform mathematical operations
✓ How to handle parentheses, exponents, and multiple operations
✓ Common mistakes to avoid
✓ Special cases like nested parentheses and implied multiplication

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