6.2 – Simplified Form for Radicals Product Rule for Square Roots Examples:

6.2 – Simplified Form for Radicals Quotient Rule for Square Roots Examples:

6.2 – Simplified Form for Radicals

6.2 – Simplified Form for Radicals Rationalizing the Denominator Radical expressions, at times, are easier to work with if the denominator does not contain a radical. The process to clear the denominator of all radicals is referred to as rationalizing the denominator

6.2 – Simplified Form for Radicals Examples:

6.2 – Simplified Form for Radicals Examples:

6.2 – Simplified Form for Radicals Theorem: If “a” is a real number, then 𝑎 2 = 𝑎 . Examples: 40 𝑥 2 𝑥 2 −16𝑥+64 18𝑥 3 −9 𝑥 2 4∙10 𝑥 2 𝑥−8 2 9𝑥 2 2𝑥−1 2 𝑥 10 𝑥−8 3 𝑥 2𝑥−1

6.3 - Addition and Subtraction of Radical Expressions Review and Examples:

6.3 - Addition and Subtraction of Radical Expressions Simplifying Radicals Prior to Adding or Subtracting

6.3 - Addition and Subtraction of Radical Expressions Simplifying Radicals Prior to Adding or Subtracting

6.3 - Addition and Subtraction of Radical Expressions Simplifying Radicals Prior to Adding or Subtracting

6.3 - Addition and Subtraction of Radical Expressions Examples:

6.3 - Addition and Subtraction of Radical Expressions Examples:

6.4 –Multiplication and Division of Radical Expressions Examples:

6.4 –Multiplication and Division of Radical Expressions Examples: 𝑥− 3𝑥 + 5𝑥 − 15

6.4 –Multiplication and Division of Radical Expressions Review: (x + 3)(x – 3)  x2 – 3x + 3x – 9  x2 – 9 𝑥 +3 𝑥 −3 𝑥 2 −3 𝑥 +3 𝑥 −9 𝑥−9

6.4 –Multiplication and Division of Radical Expressions Examples:

6.4 –Multiplication and Division of Radical Expressions If the denominator contains a radical and it is not a monomial term, then the use of a conjugate is required in order to rationalize the denominator. conjugate

6.4 –Multiplication and Division of Radical Expressions Example:

6.4 –Multiplication and Division of Radical Expressions Example:

6.5 – Equations Involving Radicals Radical Equations: The Squaring Property of Equality: Examples:

6.5 – Equations Involving Radicals Suggested Guidelines: 1) Isolate the radical to one side of the equation. 2) Square both sides of the equation. 3) Simplify both sides of the equation. 4) Solve for the variable. 5) Check all solutions in the original equation.

6.5 – Equations Involving Radicals

6.5 – Equations Involving Radicals

6.5 – Equations Involving Radicals no solution

6.5 – Equations Involving Radicals

6.5 – Equations Involving Radicals

6.5 – Equations Involving Radicals

6.5 – Equations Involving Radicals

6.6 – Complex Numbers Complex Number System: This system of numbers consists of the set of real numbers and the set of imaginary numbers. Imaginary Unit: The imaginary unit is called i, where and Square roots of a negative number can be written in terms of i.          

Operations with Imaginary Numbers 6.6 – Complex Numbers The imaginary unit is called i, where and Operations with Imaginary Numbers             

6.6 – Complex Numbers The imaginary unit is called i, where and Numbers that can written in the form a + bi, where a and b are real numbers. 3 + 5i 8 – 9i –13 + i The Sum or Difference of Complex Numbers     

6.6 – Complex Numbers      

Multiplying Complex Numbers          

Multiplying Complex Numbers         

Dividing Complex Numbers Rationalizing the Denominator:          

Dividing Complex Numbers Complex Conjugates: The complex numbers (a + bi) and (a – bi) are complex conjugates of each other and, (a + bi)(a – bi) = a2 + b2    

Dividing Complex Numbers Complex Conjugates: The complex numbers (a + bi) and (a – bi) are complex conjugates of each other and, (a + bi)(a – bi) = a2 + b2     

Dividing Complex Numbers Complex Conjugates: The complex numbers (a + bi) and (a – bi) are complex conjugates of each other and, (a + bi)(a – bi) = a2 + b2    

6.3 - Addition and Subtraction of Radical Expressions A Challenging Example 2 𝑥 2 𝑦 4 𝑧 3 1 5 2 𝑥 2 𝑧 3 2 1 5 𝑥 2 5 𝑦 4 5 𝑧 3 5 2 𝑥 2 𝑧 3 2 6 5 𝑥 12 5 𝑦 4 5 𝑧 18 5