Simplifying Square Roots

Introduction

• Focus on simplifying square roots from basic to more complex examples.

Basic Simplifications

Square Root of Perfect Squares

• Square Root of 49:
  • 49 is 7 x 7.
  • Square root is 7.
• Square Root of Negative Numbers:
  • Example: ( \sqrt{-25} )
  • Break it into 25 x (-1).
  • ( \sqrt{-1} = i ) (imaginary number).
  • ( \sqrt{25} = 5 ), so ( \sqrt{-25} = 5i ).

Real Numbers with Negative Signs Outside

• Negative Square Root of 81:
  • ( -\sqrt{81} )
  • 81 is 9 x 9, ( \sqrt{81} = 9 ).
  • Result: -9.

Negative Inside the Square Root

• Negative Square Root of Negative 64:
  • ( -\sqrt{-64} )
  • ( \sqrt{-1} = i ), ( \sqrt{64} = 8 ).
  • Result: -8i.

Simplifying Non-Perfect Squares

Concept of Perfect Squares

• Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
• Simplify non-perfect square roots by identifying the highest perfect square factor.

Examples

• Square Root of 18:
  • 18 = 9 x 2.
  • ( \sqrt{9} = 3 ), so ( \sqrt{18} = 3\sqrt{2} ).
• Square Root of 75:
  • 75 = 25 x 3.
  • ( \sqrt{25} = 5 ), so ( \sqrt{75} = 5\sqrt{3} ).

Practice Problems

• Square Root of 12:
  • 12 = 4 x 3.
  • ( \sqrt{4} = 2 ), so ( \sqrt{12} = 2\sqrt{3} ).
• Square Root of 48:
  • 48 = 16 x 3.
  • ( \sqrt{16} = 4 ), so ( \sqrt{48} = 4\sqrt{3} ).

Complex Expressions

Expression Simplification

• Expression: ( 3\sqrt{18} + 5\sqrt{72} - 4\sqrt{32} )
  • Simplify each term:
    • ( 18 = 9 \times 2 \rightarrow 3\times3\sqrt{2} = 9\sqrt{2} )
    • ( 72 = 36 \times 2 \rightarrow 5\times6\sqrt{2} = 30\sqrt{2} )
    • ( 32 = 16 \times 2 \rightarrow 4\times4\sqrt{2} = 16\sqrt{2} )
  • Combine: ( 9 + 30 - 16 = 23 )
  • Final Result: ( 23\sqrt{2} )

Conclusion

• Simplification involves identifying and employing perfect squares.
• Combine like terms when radicals are the same.
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