Lecture Notes: Multiplying Radical Expressions

Key Concepts

• Radical Expressions: An expression that contains a square root, cube root, etc.
• Index Number: The small number outside and to the left of the radical sign which indicates the degree of the root.

Basic Multiplication of Radicals

• Rule: If radicals have the same index, multiply the numbers inside the radicals.
  • Example: ( \sqrt{3} \times \sqrt{5} = \sqrt{15} )

Simplifying Radicals After Multiplication

1. Example: ( \sqrt{6} \times \sqrt{15} )

   • Multiply: ( 6 \times 15 = 90 )
   • Simplify: ( \sqrt{90} )
     • Break down: ( 90 = 9 \times 10 )
     • ( \sqrt{9} = 3 )
     • Result: ( 3\sqrt{10} )

2. Example: ( \sqrt{8} \times \sqrt{6} )

   • Multiply: ( 8 \times 6 = 48 )
   • Simplify: ( \sqrt{48} )
     • Break down: ( 48 = 16 \times 3 )
     • ( \sqrt{16} = 4 )
     • Result: ( 4\sqrt{3} )

Simplifying Before Multiplication

• Useful for large numbers.
• Example: ( 150 \times 30 )
  • Simplify: ( 150 = 15 \times 10 ), ( 30 = 15 \times 2 )
  • ( 15 \times 15 = 225 )
  • ( \sqrt{225} = 15 )
  • ( 2 \times 10 = 20 ) (( \sqrt{20} = 2\sqrt{5} ))
  • Result: ( 30\sqrt{5} )

Cube Roots

1. Example: Cube root of ( 18 \times 6 )
   • Multiply: ( 18 \times 6 = 108 )
   • Simplify: ( 108 = 27 \times 4 )
   • ( \text{Cube root of } 27 = 3 )
   • Result: ( 3\sqrt[3]{4} )

Radicals with Variables

• Example: ( \sqrt{18x^3} \times \sqrt{72x^5} )
  • ( 18 = 9 \times 2 ), ( 72 = 36 \times 2 )
  • ( x^3 \times x^5 = x^8 )
  • ( 9 = 3^2 ), ( 36 = 6^2 )
  • Result: ( 36x^4 )

Advanced Expressions

• Fourth root and cube root expressions
  • Example: Fourth root of ( 18a^{11}b^{15} \times 27a^{13}b^9 )
    • Simplify using factorization and extracting multiples of four.
    • Result: ( 3a^6b^6\sqrt[4]{6} )

Special Cases

• Perfect Square Trinomials
  • Example: ( \sqrt{x^2 + 6x + 9} )
    • Recognize as ((x+3)^2)
    • Result: ( x + 3 )

Summary

• Multiplication of radical expressions involves either directly multiplying the numbers within the radicals or simplifying before multiplication.
• Always look for possible simplifications using factorization of the numbers under the radical.
• Use perfect squares or cubes to make simplifications easier.