Lecture on Rationalizing the Denominator

Introduction

• Objective: Learn how to rationalize the denominator of a fraction.
• Key Concept: Rationalizing the denominator means converting the denominator into a rational number.
  • Irrational Numbers: Terms like roots (√) are irrational.
  • Goal: Remove irrational numbers (thirds) from the denominator.

Rationalizing Single-Term Denominators

• Example: ( \frac{4}{\sqrt{5}} )
  • Method: Multiply both numerator and denominator by the irrational denominator.
  • Steps:
    • Multiply ( \frac{4}{\sqrt{5}} ) by ( \frac{\sqrt{5}}{\sqrt{5}} ).
    • Result: ( \frac{4\sqrt{5}}{5} ).

Rationalizing Two-Term Denominators

• Example: ( \frac{7}{2 + \sqrt{3}} )
  • Method: Multiply both numerator and denominator by the conjugate of the denominator.
  • Steps:
    • Conjugate of (2 + \sqrt{3}) is (2 - \sqrt{3}).
    • Multiply ( \frac{7}{2 + \sqrt{3}} ) by ( \frac{2 - \sqrt{3}}{2 - \sqrt{3}} ).
    • Numerator: (7 \times 2 = 14) and (7 \times -\sqrt{3} = -7\sqrt{3}).
    • Denominator:
      • (2 \times 2 = 4)
      • (2 \times -\sqrt{3} = -2\sqrt{3})
      • (\sqrt{3} \times 2 = 2\sqrt{3})
      • (\sqrt{3} \times -\sqrt{3} = -3)
    • Simplified Denominator: (4 - 3 = 1)
    • Result: (14 - 7\sqrt{3})

Additional Examples

Example 1: Rationalizing Single-Term Denominator

• Problem: ( \frac{6}{\sqrt{3}} )
• Solution:
  • Multiply by ( \frac{\sqrt{3}}{\sqrt{3}} ).
  • Results to ( \frac{6\sqrt{3}}{3} ).
  • Simplify by dividing both terms by 3 to get (2\sqrt{3}).

Example 2: Form Requirement with Two-Term Denominator

• Problem: ( \frac{7 + \sqrt{5}}{\sqrt{5} - 1} )
• Form Requirement: (a + b\sqrt{5}) where (a, b) are positive integers.
• Solution:
  • Multiply by ( \frac{\sqrt{5} + 1}{\sqrt{5} + 1} ).
  • Numerator calculation yields (12 + 8\sqrt{5}).
  • Denominator simplifies to 4 after cancelling terms.
  • Final Simplified Result: (3 + 2\sqrt{5})

Conclusion

• Rationalizing involves removing irrational numbers from the denominator.
• Apply appropriate multiplication techniques based on the form of the denominator.
• Ensure final result meets any specified form requirements.