Lecture on Adding and Subtracting Radicals

Introduction

• Focus: Understanding how and when to add and subtract radicals.
• Goal: Provide a foundation for combining equations and expressions involving radicals.

Basics of Addition and Subtraction

• Example: 5 + 5 = 10
  • Alternate approach: 5 + 5 can be represented as 2 times 5.
  • Important concept: Grouping similar terms for simplification.
• Non-similar numbers:
  • 5 + 6 = 11 cannot be grouped like 2 times 5 or 2 times 6.
  • Analogy: Adding oranges and apples cannot be simply combined as fruit.
  • Variables:
    • x + x = 2x by grouping.
    • x + y cannot be combined further.
    • x² + x² = 2x² can be grouped.
    • x² + y² cannot be combined.

Combining Radicals

• Like Terms in Radicals:
  • √x + √x = 2√x when the radicands and indices are the same.
  • √x + √y or √x + ³√x cannot be combined due to different radicands/indices.
• Key Rule:
  • Elements to be combined must be exactly the same in both radicands and indices.

Example 1: Simplifying Radicals

• Problem: √18 + √32
  • Indices are the same (both are square roots).
  • Different radicands require simplifying.
• Simplification Process:
  • √18 = √(9×2) and √32 = √(16×2)
  • Simplify to 3√2 + 4√2
  • Combine to get 7√2 as the radicands and roots are the same.

Example 2: Cube Roots

• Problem: Involves cube roots with numbers in front.
  • Ensure indices are the same (both are cube roots).
  • Simplify to find common radicands.
• Simplification Process:
  • Recognize largest cube number in 40 as 8 (8 and 27 are base cubes).
  • Decompose: ³√40 = ³√(8×5)
  • Simplify to 7³√5 - 8³√5
  • Combine to get -³√5 as final result.

Conclusion

• Importance of identical indices and radicands for combining radicals.
• Additional resources: Check further examples and videos for deeper understanding.

This lecture covered the foundational principles and specific examples of adding and subtracting radicals, emphasizing the need for identical radicands and indices to successfully combine them.