Order of Operations (PEMDAS)

Introduction

• PEMDAS: Acronym for the order of operations in mathematics.
  • P: Parentheses
  • E: Exponents
  • M: Multiplication
  • D: Division
  • A: Addition
  • S: Subtraction
• Priority: Parentheses > Exponents > Multiplication/Division > Addition/Subtraction.

Order of Operations

Parentheses

• Resolve expressions within parentheses first.

Exponents

• Calculate exponents after dealing with parentheses.

Multiplication and Division

• Perform from left to right.
• Example: 18 ÷ 3 × 4
  • Division first: 18 ÷ 3 = 6; 6 × 4 = 24 (Correct)
  • Multiplication first: 3 × 4 = 12; 18 ÷ 12 = 1.5 (Incorrect)

Addition and Subtraction

• Similar priority, but execute from left to right.
• Example: 5 + 8 - 3
  • 5 + 8 = 13; 13 - 3 = 10
  • 8 - 3 = 5; 5 + 5 = 10 (Same result in both cases)

Examples

Mixed Operations

• Example: 8 + 9 × 6
  • Multiply first: 9 × 6 = 54; 8 + 54 = 62 (Correct)
  • Add first: 8 + 9 = 17; 17 × 6 = 102 (Incorrect)

Division First

• Example: 16 - 14 ÷ 2
  • Divide first: 14 ÷ 2 = 7; 16 - 7 = 9

Combined Multiplication/Division

• Work from left to right.
  • Example 1: 48 ÷ 6 × 2 + 7
    • 48 ÷ 6 = 8; 8 × 2 = 16; 16 + 7 = 23
  • Example 2: 3 × 12 ÷ 4 - 9
    • 3 × 12 = 36; 36 ÷ 4 = 9; 9 - 9 = 0

Exponents

Calculating Exponents

• Example: 3²
  • 3 × 3 = 9

Negative Numbers and Exponents

• Example: (-3)² vs -3²
  • (-3)² = (-3) × (-3) = 9
  • -3² = -(3 × 3) = -9

Fractions and Exponents

Evaluating Expressions

• Example 1: 3 × 4² - 3² × 5 ÷ (5 × 9)

  • Top: 3 × 16 = 48; 9 × 5 = 45; 48 - 45 = 3
  • Bottom: 45; Simplify 3/45 = 1/15

• Example 2: 3 + 4² ÷ 2² × 5

  • Top: 4² = 16; 3 + 16 = 19
  • Bottom: 2² = 4; 4 × 5 = 20; Result = 19/20

• Example 3: 5 × 8 × 6² ÷ (12 × 10 × 4)

  • Break down and cancel similar factors
  • Simplified result: 3

Conclusion

• Key Rule: Follow PEMDAS accurately to solve mathematical expressions correctly.
• Tips: Always resolve operations from left to right within the same priority level.