1. 7.5 – Rationalizing the Denominator of Radicals Expressions
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 radical is referred to as rationalizing the denominator

2. 7.5 – Rationalizing the Denominator of Radicals Expressions

3. 7.5 – Rationalizing the Denominator of Radicals Expressions

4. 7.5 – Rationalizing the Denominator of Radicals Expressions
If the denominator contains a radical and it is not a monomial term, then the use of a conjugate is required.
Review:
(x + 3)(x – 3) 
x2 – 3x + 3x – 9 
x2 – 9
(x + 7)(x – 7) 
x2 – 7x + 7x – 49 
x2 – 49

5. 7.5 – Rationalizing the Denominator of Radicals Expressions
If the denominator contains a radical and it is not a monomial term, then the use of a conjugate is required.
conjugate

6. 7.5 – Rationalizing the Denominator of Radicals Expressions
conjugate

7. 7.5 – Rationalizing the Denominator of Radicals Expressions
conjugate

8. 7.6 – Radical Equations and Problem Solving
The Squaring Property of Equality:
Examples:

9. 7.6 – Radical Equations and Problem Solving
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.

10. 7.6 – Radical Equations and Problem Solving

11. 7.6 – Radical Equations and Problem Solving

12. 7.6 – Radical Equations and Problem Solving
no solution

13. 7.6 – Radical Equations and Problem Solving

14. 7.6 – Radical Equations and Problem Solving

15. 7.6 – Radical Equations and Problem Solving

16. 7.6 – Radical Equations and Problem Solving

17. 7.7 – 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.        

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

19. 7.7 – 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     

20. 7.7 – Complex Numbers      

21. Multiplying Complex Numbers       

22. Multiplying Complex Numbers      

23. 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    

24. 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    

25. 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    