Algebra Lesson: Exponents and the Laws of Exponents

Introduction

• Instructor: Mr. Cosi
• Location: Atascosita, Texas
• Objective: Understanding exponents and laws of exponents
• Prerequisites:
  • Knowledge of math facts and vocabulary (e.g. sum, difference, product)
  • Understanding of fractions

Exponents

• Definition: Indicates how many times a number (base) is a factor.
  • Example: (x^m) means the base is (x) and it is multiplied by itself (m) times.
• Terminology: Exponents are also called 'powers'.
• Examples:
  • (x^2) or (x) squared
  • (x^3) or (x) cubed
  • (b^4)

Combining Exponents

• Like Terms: Must add or subtract like terms (same base and exponent).
  • Example: (x^m + x^n + x^m = 2x^m + x^n)

Laws of Exponents

1. Product of Powers

• Rule: When multiplying, add the exponents.
  • Example: (x^2 \times x^3 = x^{2+3} = x^5)

2. Power of a Product

• Rule: Apply power to each factor and simplify.
  • Example: ((3ab)^2 = 3^2 \times a^2 \times b^2 = 9a^2b^2)

3. Power of a Power

• Rule: When raising a power to another power, multiply the exponents.
  • Example: ((x^2)^3 = x^{2 \times 3} = x^6)

4. Quotient of Powers

• Rule: When dividing, subtract the exponents.
  • Example: (x^3 / x^2 = x^{3-2} = x^1 = x)

5. Power of a Quotient

• Rule: Apply the power to the numerator and the denominator.
  • Example: ((\frac{2a}{b})^3 = \frac{2^3 \times a^3}{b^3} = \frac{8a^3}{b^3})

6. Zero Exponent

• Rule: Any number raised to the power of zero equals one.
  • Example: (x^0 = 1), ((xy)^0 = 1)
  • Proof: (x^3 / x^3 = x^{3-3} = x^0 = 1)

7. Negative Exponents

• Rule: A negative exponent indicates a reciprocal.
  • Example: (a^{-2} = \frac{1}{a^2}) and (\frac{1}{a^{-2}} = a^2)

Important Considerations

• Denominators cannot be zero (cannot divide by zero).

Recap

• Defined exponents
• Stated and demonstrated the laws of exponents
• Encouraged to reach out via email for questions

Contact

• Email: [email protected]
• Closing: Have a great day!