Multiplying and Dividing Radical Expressions

Key Concepts

• Radicand: The number inside the square root symbol.
• Multiplying Radicals: Multiply the radicands, put the product under a single square root, and then simplify.
• Dividing Radicals: Simplify the fraction under a single square root and then rationalize the denominator if necessary.

Multiplying Radical Expressions

1. Multiply Radicands

   • Example: ( 3 \sqrt{6x} \times \sqrt{10x} )
     • Multiply (6x) and (10x) to get (60x^2).
     • Expression becomes (3\sqrt{60x^2}).

2. Simplify

   • Find perfect square factors within the radicand.
   • Example: (60) has a perfect square factor of (4) ((4 \times 15)).
   • Break into (\sqrt{4}\times\sqrt{15}\times\sqrt{x^2}).
   • Simplify: (\sqrt{4} = 2) and (\sqrt{x^2} = x).
   • Result: (6x\sqrt{15}).

FOIL Method with Radicals

• Apply FOIL when multiplying binomials, whether numbers or radicals:
  • Example: ((4 + \sqrt{5})(1 - \sqrt{5}))
  • FOIL: First, Outer, Inner, Last.
  • Simplify like terms and radicals:
    • Combine like radicals: (-4\sqrt{5} + \sqrt{5} = -3\sqrt{5}).
    • (\sqrt{5} \times \sqrt{5} = \sqrt{25} = 5).
    • Simplified expression: (-1 - 3\sqrt{5}).

Dividing Radical Expressions

1. Express as a Fraction

   • (\frac{\sqrt{2x}}{\sqrt{6}} = \sqrt{\frac{2x}{6}} = \sqrt{\frac{1x}{3}})
   • Simplify radicand if possible.

2. Rationalize the Denominator

   • If a radical remains in the denominator:
     • Multiply numerator and denominator by the radical in the denominator.
     • Example: (\frac{1}{\sqrt{3}} \to \frac{\sqrt{3}}{3}).

3. Alternate Method

   • Simplify radicands directly and then rationalize.
   • Example: (\frac{\sqrt{2}}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{\sqrt{12}}{6}).
   • Simplify further: (\sqrt{12} = \sqrt{4\times3} = 2\sqrt{3}).
   • Result: (\frac{2\sqrt{3}}{6} = \frac{\sqrt{3}}{3}).

Important Notes

• Always look for and simplify perfect square factors.
• Rationalize denominators to eliminate radicals in the denominator.
• Continually simplify expressions for clarity and simplicity.

Conclusion

• Multiplying and dividing radical expressions involves simplifying and rationalizing.
• Use methods like FOIL and rationalization.
• Practice helps improve skills in handling radical expressions effectively.