Simplifying Radicals

Introduction

• Focus on simplifying radicals including the square roots and dealing with negative and imaginary numbers.

Simple Square Roots

• Square Root of 36: ( \sqrt{36} = 6 ) (a real number).

Square Roots of Negative Numbers

• Square Root of -49: ( \sqrt{-49} = 7i )
  • ( i ) is the imaginary unit, ( i = \sqrt{-1} ).
  • Separate into ( \sqrt{49} ) and ( \sqrt{-1} ) to get ( 7i ).
• Negative Square Root of 64: ( -\sqrt{64} = -8 ).
• Negative Square Root of -25: ( -\sqrt{-25} = -5i ).

Understanding Perfect Squares

• List of Perfect Squares (1 to 10):
  • 1² = 1
  • 2² = 4
  • 3² = 9
  • 4² = 16
  • 5² = 25
  • 6² = 36
  • 7² = 49
  • 8² = 64
  • 9² = 81
  • 10² = 100
• Beyond 10:
  • 20² = 400

Simplifying Non-Perfect Squares

• Square Root of 75: Break down into 25 and 3.
  • ( \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} ).
• Square Root of 18: ( \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} ).
• Square Root of 48: ( \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} ).

Practice Problems

• 8( \sqrt{80} ): ( \sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5} ); Result: 32( \sqrt{5} ).
• 5( \sqrt{98} ): ( \sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2} ); Result: 35( \sqrt{2} ).

Rationalizing Denominators

• ( \frac{5}{\sqrt{2}} ): Multiply top and bottom by ( \sqrt{2} ).
  • Result: ( \frac{5\sqrt{2}}{2} ).

Adding and Subtracting Radicals

• Examples of simplifying combined expressions by breaking down into perfect squares.
• Expression: 4( \sqrt{8} ) + 3( \sqrt{15} ) - 6( \sqrt{32} )
  • Simplified Form: (-\sqrt{2} ).

Advanced Examples

• Expression: 7( \sqrt{27} ) + 3( \sqrt{12} ) - 5( \sqrt{48} )
  • Simplified Form: 7( \sqrt{3} ).

Simplifying Radical Fractions

• Expression: ( \frac{8}{3\sqrt{2}} )

  • Multiply by conjugate.
  • Result: ( \frac{24 + 8\sqrt{2}}{7} ).

• Expression: ( \frac{3 + \sqrt{2}}{5 - \sqrt{2}} )

  • Multiply by conjugate.
  • Result: ( \frac{17 + 8\sqrt{2}}{23} ).

Conclusion

• Simplifying radicals involves identifying perfect squares, manipulating imaginary units, and rationalizing denominators.
• Practice with examples to master the technique of simplifying radical expressions.