2. 6.2 – Simplified Form for Radicals
Product Rule for Square Roots
Examples:

3. 6.2 – Simplified Form for Radicals
Quotient Rule for Square Roots
Examples:

4. 6.2 – Simplified Form for Radicals

5. 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. 6.2 – Simplified Form for Radicals
Examples:

7. 6.2 – Simplified Form for Radicals
Examples:

8. 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

9. 6.3 - Addition and Subtraction of Radical Expressions
Review and Examples:

10. 6.3 - Addition and Subtraction of Radical Expressions
Simplifying Radicals Prior to Adding or Subtracting

11. 6.3 - Addition and Subtraction of Radical Expressions
Simplifying Radicals Prior to Adding or Subtracting

12. 6.3 - Addition and Subtraction of Radical Expressions
Simplifying Radicals Prior to Adding or Subtracting

13. 6.3 - Addition and Subtraction of Radical Expressions
Examples:

14. 6.3 - Addition and Subtraction of Radical Expressions
Examples:

15. 6.4 –Multiplication and Division of Radical Expressions
Examples:

16. 6.4 –Multiplication and Division of Radical Expressions
Examples:
𝑥− 3𝑥 + 5𝑥 − 15

17. 6.4 –Multiplication and Division of Radical Expressions
Review:
(x + 3)(x – 3) 
x2 – 3x + 3x – 9 
x2 – 9
𝑥 𝑥 −3
𝑥 2 −3 𝑥 +3 𝑥 −9
𝑥−9

18. 6.4 –Multiplication and Division of Radical Expressions
Examples:

19. 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

20. 6.4 –Multiplication and Division of Radical Expressions
Example:

21. 6.4 –Multiplication and Division of Radical Expressions
Example:

23. 6.5 – Equations Involving Radicals
Radical Equations:
The Squaring Property of Equality:
Examples:

24. 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.

25. 6.5 – Equations Involving Radicals

26. 6.5 – Equations Involving Radicals

27. 6.5 – Equations Involving Radicals
no solution

28. 6.5 – Equations Involving Radicals

29. 6.5 – Equations Involving Radicals

30. 6.5 – Equations Involving Radicals

31. 6.5 – Equations Involving Radicals

33. 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.          

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

35. 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     

36. 6.6 – Complex Numbers      

37. Multiplying Complex Numbers         

38. Multiplying Complex Numbers        

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

40. 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    

41. 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     

42. 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    

44. 6.3 - Addition and Subtraction of Radical Expressions
A Challenging Example
2 𝑥 2 𝑦 4 𝑧
2 𝑥 2 𝑧 3
𝑥 𝑦 𝑧 3 5
2 𝑥 2 𝑧 3
𝑥 𝑦 𝑧 18 5