Lecture Notes: Section 6.2 - Multiplying and Dividing Radicals

Overview

• Focus on multiplying and dividing radicals, building on 6.1
• Importance of being able to simplify radicals and complete the related homework

Multiplying Radicals

Example 1

• Method 1:

  • Take square root of each factor (e.g., (\sqrt{4} = 2) and (\sqrt{25} = 5)), then multiply results: (2 \times 5 = 10)

• Method 2:

  • Multiply under the radical first ((4 \times 25 = 100)) and then take square root: (\sqrt{100} = 10)

Example 5

• Separate Multiples:
  • For non-perfect squares (e.g., (\sqrt{18})), create a factor tree:
    • (18 \to 2, 9 \to 3, 3) (\Rightarrow) One pair of threes, so (3) comes out, (2) stays under radical
  • (\sqrt{50} \to 2, 25 \to 5, 5) (\Rightarrow) One pair of fives, so (5) comes out, (2) stays under radical
  • Multiply results: (3 \times 5 = 15); (15 \times 2 = 30)

Calculator Use

• Calculators can simplify if they yield a whole number; otherwise, factor trees are needed.

Variables and Different Indices

Example 8

• Cubic Roots:
  • (\sqrt[3]{125} = 5)
  • (Y^2) cannot fully resolve into groups of three, stays as (\sqrt[3]{Y^2})
  • (Z^4) yields one pair of three ((Z) outside) and (Z^1) remains under radical
  • Index must be noted when not (2)

Example 11

• Standard Roots (index = 2):
  • (\sqrt{\text{no index, assume 2}}) (\Rightarrow) Work with pairs of twos

Example 12 and 13

• Fourth and Cubic Roots with Indices:
  • (\sqrt[4]{256} = 4)
  • Pairing rules apply based on indices

Example 19

• Multiply coefficients and terms under radicals, simplify using known square roots

Dividing Radicals

Example 26

• Basic Division:
  • Divide coefficients; subtract exponents for variables
  • Simplify using square roots

Example 27

• Factor Trees for Imperfect Squares:
  • Factorize, pair, and simplify

Rationalizing the Denominator

Key Concept

• Cannot have a square root in the denominator
• Multiply numerator and denominator by (\sqrt{\text{denominator}}) to rationalize

Homework

• Review notes thoroughly
• Homework will include these concepts; contact if questions arise

Closing

• Homework and additional practice to be shared
• Contact for questions or clarification