Lecture Notes: Exponents and Order of Operations

Introduction to Exponents

• Concept of Repeated Operations
  • Addition: Repeating addition results in multiplication.
  • Multiplication: Repeating multiplication introduces the concept of exponents.
• Defining Exponents
  • Base: The number being multiplied repeatedly.
  • Exponent: The number of times the base is multiplied by itself.
  • Example: (5^7) means 5 is multiplied by itself 7 times.

Terminology

• Square: (5^2) is expressed as "5 squared."
• Cube: (5^3) is referred to as "5 cubed."
• General form: "5 to the power of n" where n is any number.

Evaluating Exponents

• Example: (7^3 = 343) not 21.
• First Power: Any number to the first power is itself (e.g., (11^1 = 11)).

Introduction to Order of Operations

• Purpose: Ensures consistent mathematical results.
• Mnemonic: PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).

Detailed Explanation of Order of Operations

• Step 1: Parentheses
  • Perform all operations inside parentheses first.
  • Different types of brackets are all considered parentheses.
• Step 2: Exponents
  • Evaluate any terms with exponents next.
• Step 3: Multiplication and Division
  • Perform from left to right as they appear.
• Step 4: Addition and Subtraction
  • Also performed from left to right.

Applying Order of Operations

• Examples
  • (8 + 3 \times 5): Calculate multiplication first.
  • (48 \div 3 \times 4): Perform division before multiplication.

Advanced Problems and Fractional Calculations

• Parentheses and Nested Operations
  • Solve innermost parentheses first followed by outer brackets.
  • Example: (4 + (9 - 5)^2 - 7 \times 3)
• Fractions as Division
  • Treat large fractions as implied parentheses.
  • Solve numerator and denominator separately before dividing.
  • Example: (\frac{25 + 8 \times 2 - 3^3}{5 \times (2 - 1)})

Area of a Square

• Definition: All sides are equal in a square.
• Calculating Area: Side squared (e.g., if side = 8, area = (8^2 = 64) square inches).
• Perimeter: Sum of all sides (e.g., if side = 8, perimeter = 4 \times 8 = 32 inches).

Final Notes

• Understanding Exponents and Order of Operations is crucial for solving complex expressions.
• Homework: Practice problems on order of operations and areas as given.
• Resources: Utilize online resources and math labs for additional help.

This concludes the notes on exponents and order of operations. Make sure to practice and reach out for help if needed.