Simplifying Fractions and Exponents

Oct 6, 2024

Algebra 1: Unit 6, Lesson 4 - Simplifying Fractions with Exponents

Introduction

  • Instructor: Kirk Weiler
  • Focus: Simplifying fractions involving exponents

Key Concepts

Exponent Laws

  • We previously learned two exponent laws.
  • Today: Develop a third law involving division of bases with exponents.

Fundamentals of Fractions and Division

  • Division and fractions are closely related.
  • Key idea: Any non-zero quantity divided by itself equals 1.
  • Simplifying fractions: Numerator and denominator are the same.

Exercise 1: Simplifying Basic Fractions

  • Example A: 7/7 = 1
  • Example B: 5^2 / 5^2 = 1
  • Example C: n/n = 1
  • Example D: (10x^3)/(10x^3) = 1

Exercise 2: Simplifying Numerical Fractions

  • Fraction: 18/30
  • GCF: 6 (Largest number dividing both 18 and 30)
  • Factored Form: 18 = 3x6, 30 = 5x6
  • Simplification: (3/5) * (6/6) = 3/5

Multiplying Fractions

  • Multiply numerators and denominators.
  • Un-multiplication: Break into products.

Exercise 3: Fractions with Variables and Exponents

  • Fraction: 2x^5 / 6x^3
  • GCF: 2x^3
  • Simplification Process:
    • Write in factored form: (2x^3 * x^2) / (2x^3 * 3)
    • Result: x^2 / 3

Exercise 4: Practice with Tricky Cases

  • Example A: x^2 / x^6 = 1 / x^4
  • Example B: 10m^9 / 2m^4 = 5m^5
  • Example C: 6y^3 / 4y^7 = 3 / 2y^4
  • Example D: 4x^3 / 12x^8 = 1 / 3x^5

Key Point

  • Always have a '1' in the numerator when everything cancels out.
  • Never assume remaining terms are whole numbers unless explicitly stated.

Exercise 5: Simplifying Variable Fractions

  • x^a / x^b simplifies based on exponent difference.
  • If numerator power > denominator power:
    • Result is in the numerator (e.g., x^(a-b)).
  • If denominator power > numerator power:
    • Result is in the denominator (e.g., 1/x^(b-a)).

General Patterns

  • Numerator Power > Denominator Power: x^a / x^b = x^(a-b)
  • Denominator Power > Numerator Power: x^a / x^b = 1 / x^(b-a)

Exercise 7: Applying Patterns

  • Example A: x^8 / x^5 = x^3
  • Example B: y^4 / y^6 = 1 / y^2
  • Example C: 8t^7 / 12t^4 = 2t^3 / 3
  • Example D: 15r / 3r^5 = 5 / r^4

Conclusion

  • Simplification processes use exponent laws and fraction properties.
  • Recognize and utilize patterns for efficiency.
  • Next Lesson: Developing the third exponent law.