Lecture on Radical Expressions

Key Concepts

• Radical Expressions: Involve operations like addition, subtraction, and multiplication.
• Radicand: The expression inside the square root.

Adding and Subtracting Radical Expressions

• Like Terms: Radicals with the same radicand can be combined.
  • Example: ( 6\sqrt{5} + 2\sqrt{5} - 5\sqrt{5} ) results in ( 3\sqrt{5} ).
  • Treat radicals like variables (e.g., ( x )).
• Example: ( 7\sqrt{2} - 6\sqrt{2} = \sqrt{2} ).
• Example: ( 8\sqrt{11} - 4\sqrt{11} = 4\sqrt{11} ).
• Different Radicands: Can't combine different radicands unless simplified.

Simplifying Radicands

• Look for perfect squares within radicands to simplify.
  • Example: ( \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3} ).
  • Simplify ( 6\sqrt{9 \times 3} ) to ( 18\sqrt{3} ).
• Similar simplifications for: ( 8\sqrt{12} ) and ( 2\sqrt{75} ).

Multiplying Radical Expressions

• Multiply Coefficients and Radicals Separately:
  • Example: ( 2\sqrt{3} \times 4\sqrt{6} ) becomes ( 8\sqrt{18} ).
• Simplify the product of the radicands.
  • Example: Simplify ( \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} ), leading to ( 24\sqrt{2} ).

Distributive Property

• Example: ( 4\sqrt{2}(3\sqrt{2} + \sqrt{6}) ).
  • Multiply outside coefficients and radicands separately.
  • Result: ( 12\sqrt{4} + 4\sqrt{12} ).
  • Simplification results in final forms.

FOIL Method for Binomials

• Find area of rectangle using radicals:
  • Length: ( 5\sqrt{3} + 7\sqrt{\pi} )
  • Width: ( 4\sqrt{6} - 2\sqrt{10} )
  • Use FOIL method to expand and simplify.
  • Combine like terms post-expansion.

Summary

• Operations: Addition, subtraction involves combining like terms. Multiplication involves multiplying coefficients and radicands and then simplifying.
• Simplification: Always check for possible simplifications of radicands by finding perfect squares.