Simplifying Radicals and Radical Expressions Lecture

Introduction

• Presenter: Nancy
• Topic: Simplifying radicals, specifically square roots, to their simplest form or 'simplified radical form'.
• Structure: Starting from simple examples to more complicated cases.

Simplifying Square Roots

Perfect Squares

• Definition: Numbers that have integers as their square roots (e.g., 1, 4, 9, 16, etc.).
• Process: If the number under the square root is a perfect square, replace the square root with the integer that multiplies to it.
  • Example: √16 = 4 since 4 x 4 = 16.
  • Example: √81 = 9 since 9 x 9 = 81.

Non-Perfect Squares

• Process: Find a perfect square factor of the number under the square root.

  • Example: For √32:
    1. Find factors: 32 can be divided by 4.
    2. Break down: √32 = √(4x8) = 2√8.
    3. Simplify further: √8 = √(4x2) = 2√2.
    4. Final: 4√2.

• Larger Numbers: Same principle applies.

  • Example: √125 = √(25x5) = 5√5.

Simplifying Products

• Definition: Numbers multiplied by a square root.

• Process: Simplify the square root portion.

  • Example: For 5√54:
    1. Break down 54: √54 = √(9x6) = 3√6.
    2. Combine constants: 5 x 3√6 = 15√6.

• Non-Simplifiable Roots

  • Example: 4√31 remains as is since 31 is prime.

Simplifying Quotients

• Objective: Remove square roots from the denominator.
• Process:
  • Simple Case: Multiply by the square root over itself.
    • Example: 4/√2 = 4√2/2 = 2√2.
  • Complex Case: Use the conjugate.
    • Example: (4 + √3)/(5 - √3) requires multiplying by the conjugate (5 + √3)/(5 + √3).
    • Perform distribution (FOIL) and simplify.
    • Result: 23 + 9√3 / 22.

Summary

• Key Techniques: Identifying perfect squares, using conjugates, simplifying through multiplication and division.
• Goal: Achieve the simplest radical form by removing radicals from the denominator and simplifying the expression.
• Final Note: Simpler forms are preferred in mathematical expressions.