2. WARM-UP Find the prime factorization for the following numbers. Write your answer as a product of primes. 1.722.120

3. 5.1a Rational Exponents & Simplify Radicals nn 2 n 3 n 4 n 5 n 6 n 7 n 8 248163264128256 3927812437292187 416642561024 525125625 6362161269 7493432401 864512 981729 Objective: To simplify rational exponents

4. Radicals Root index What should you do if the exponent is not even?

5. Simplify: All variables are positive. Cube roots: Look for perfect cubes in the coefficient. How can you determine if the variable is a perfect cube?

6. Rational Exponents 2 3 = 8 5 4 = 625 Now – Try some fun problems! Remember: Root first makes the number smaller. Can you simplify rational exponents? Assign 5.1a: 17-39 odd, 41-58 all

7. WARM-UP Hmmm…do you remember?? 1.x 2 x 3 = 2.(x 2 ) 3 = 3. a -3 4. =

8. 5.1a Answers 42. 16 44. 27 46. 27 48. 1/3 50. 1/8 52. 7/4 54. 4/9 56. 15 58. 12/35

9. 5.1b Simplifying Radical Expressions Recall the exponent properties. x a x b = (x a ) b = a -n = x 0 = 1, Objective: To simplify rational expressions using exponent properties

10. Now try these!

11. Simplify. All variables are positive. Can you simplify rational expressions using exponent properties? Homework: 5.1: 59-67 odd, 68-78 all, 93, 94, 107, 108 Quiz after 5,3

12. 5.1b Answers 68. 70. 72. 74. 76. 78. 94. 108.

13. 5.2 WARM-UP Simplify:

14. 5.2 More Rational Exponents Multiply: Objective: To continue multiplying rational exponents

15. Multiply: Factor completely: Ex 7. 4(x – 3) 2 + 5x(x – 3) =

16. Factor with Rational Exponents Determine the smallest exponent and factor this from all terms. Try these:

17. Last one! Add: Don’t forget the common denominator! Can you multiplying rational exponents? Assign 5.2: 3-69 (x3), 77, 81, 97-100

18. 5.2 Answers 6.12.18. 24. 30. t - 12536. a + 27 42.48.54. 60.66.98. 100.

19. 5.3 Simplified Radical Form Properties for radicals: a, b > 0 Simplify each radical means: No perfect squares left under the No perfect cubes left under the No factors in the radicand can be written as powers of the index. No fractions under the radical No radicals in the denominator Objective: To write Radicals in simplest radical form

20. When you simplified radicals to this point the book said that all variables were positive. What if they do not tell us all variables are positive? The first one needs absolute value symbols to insure the answer is positive The second does not because if x was negative, it could not be under an even root. Simplify each: Do not assume variables are positive. When you have an even root and an even exponent in the radicand that becomes an odd exponent when removed, you must use absolute value.

21. Type 1: Similar to section 5.1 Type 2: No radicals in the denominator.

22. Try these: How do you know what degree to make the exponents in the denominator? Can you write Radicals in simplest radical form? Assign 5.3: 3-21 (x3), 23-33 odd, 48-69 (x3), 71-77 odd, 85-87 all, 89, 105

23. GROUP ACTIVITY Learning Target:Find a set containing 3 equivalent forms of the same number on the face. You will work with the 1 or 2 people sitting beside you. Begin with all of the cards face-up spread out on the desk. Take turns gathering sets of 3 cards.

24. 5.3 Answers 6.12. 18.48. 54.60. 66.86.

25. Review 5-1 to 5-3 Questions? Remember NO CALCULATOR! 5.1:  Simplify radicals and rational exponents  Write radical expressions with rational exponents  Evaluate rational exponents  Simplify expressions with rational exponents 5.2: Multiply and factor using rational exponents Add by making common denominators with rational exponents 5.3:  Simplify radicals if the variables may not be positive  No fractions under the radical  No radicals in the denominator  Be able to do these for any root Now let’s try some problems!

26. Write using rational exponents: Simplify:

27. Multiply: Assign: Review WS

28. 5.4 – Addition and Subtraction of Radicals Objective: To add and subtract radicals We all know how to simplify an equation such as: 2x +3y – 5x = 3y – 3x The process for addition and subtraction of radicals is very similar. To do so you must have the same index and the same radicand. Lets try some! **You may need to simplify first!

29. Can you add and subtract radicals? Homework: 5.4

30. 5.4 Answers 54.

31. WARM-UP Simplify. What did you notice about the above? These are called CONJUGATES!

32. 5.5 Multiplication and Division of Radicals Objective: To multiply and divide radicals Recall the radical properties we learned earlier in the chapter. Then simplify if possible. Therefore factorable!!!

33. Now for division. Don’t forget to rationalize the denominator!! Multiply the numerator & denominator by the conjugate of the denominator. Then FOIL. Can you multiply and divide radicals? Homework: 5.5

34. 5.5 Answers 6.420012. 18. 24. 30. 3 36. x - 2242. 48.54.60. 66.

35. 5.6a Equations with Radicals Objective: To solve basic radical equations Recall: 4x – 5 = 23  Locate the variable.  Undo order of operations to isolate the variable. +5 +5 4x = 28 4 4 x = 7 How is similar? Procedure: Locate and isolate the radical. +5 +5 4 x = 28 4 4 x = 7 How do you undo the radical? ( ) 2 x = 49 Always check these answers. When you square, you may get extraneous roots.

36. Squaring Property: If both sides of an equation are squared, the solutions to the original equation are also solutions to the new equation. *You must square the entire side. *You must check for extraneous(extra) roots. BASIC:

37. Medium: * Isolate the radical on one side. * Square both sides (the entire side- FOIL) * Solve the quadratic. (How?) ( ) 2 x 2 – 6x + 9 = x – 3 x 2 – 7x + 12 = 0 (x – 4) (x – 3) = 0 x = 4, 3 Check both answers – one generally does not work. You try these: Can you solve basic radical equations? Assign: 5-6 to # 35

38. 5.6b Warm - up Solve:

39. 5.6b More Solving Radical Equations Objective: To solve radical equations with radicals on both sides and identiry extaneous roots What happens when you have two radicals that you cannot combine? * Two different roots & something else *Isolate the more complicated radical on one side and square both sides. (The entire side.) * Isolate the radical that is left and square both sides again.

40. You try this one:

41. Graphing x y 0 0 1 1 4 2 What is the domain: What is the range:

42. How would each change affect the graph? Give the domain and range for each. Domain: Range: Summary: Can you solve radical equations with radicals on both sides and identity extaneous roots? Assign: Rest of 5.6 How can you make the root open left? Upside down?

43. 5.6b Solutions 42.4 48.5, 13 54. 56.10 58.10,000 60.The plume would be smaller if there was a current.

44. 5.6c Solving Equations with Rational Exponents Objective: To solve equations with rational exponents and understand extraneous roots * To solve an equation with a rational exponent, you must first solve for the variable or parenthesis with the rational exponent. * You must undo the exponent, by taking it to a power that will cancel the exponent to a 1. ( ) 3 x 1 = 64

45. How do you know when you should use for your solution? When solving an equation and you must take an even root, you must use x = answer. You try these:

46. Miscellaneous Completely factor: x 2n – 5x n + 6 Now try: Ex: 2x 4n + x 2n - 6 Ex: x 2n+1 - 5x n+1 + 6x Cancel: Assignment: Worksheet and begin test review. Can you solve equations with rational exponents and understand extraneous roots?

47. 5.6c Worksheet Solutions 1.2710.19. 25 2.1620. 27 & -64 3.3211. 8121. 49 & 25 4.-3212. 102422. 25 5.6413. 63, -6223. 32,768 & -32 6.14. 34124. 9 &.25 25. 7.15.26. 8. 1416. 8127. 17. 3228. 9.18. 35 & -2929. 30. 31.

48. 5.7a Introduction to Complex Numbers Objective: To define imaginary and complex numbers and perform simple operations on each Real Numbers (R): the set of rational and irrational numbers. Rational (Q) : any number that can be written as a fraction Irrational (Ir) : non-repeating, non-terminating decimals Integers (I or Z) : positive and negative Whole numbers: no fractions or decimals Whole (W) : {0, 1, 2, 3, …} no fractions, decimals or negatives. Natural (N) : counting numbers no decimals, fractions, negatives, 0 Complex Numbers (C) : a + bi a = real part b = imaginary part Imaginary (Im ): What are imaginary numbers? Symbol? Value? Square roots of negative numbers. i

49. A complex number is in the form of a + bi where a = real part and bi = imaginary part. A pure imaginary number only has the imaginary part, bi. **Always remove the negative from the radical first!** Ex1: Ex2:Ex3: Ex4:Ex5:Ex6:

50. What is the value of i 2 ? ( ) 2 -1 = i 2 When you get an i 2, always replace it with a -1. i = i i 2 = -1 i 3 = i 4 = i 5 = i 6 = i 7 = i 8 = i 9 = i 10 = i 11 = i 12 = Ex1: i 20 = Ex2: i 30 = Ex3: i 57 = Ex4: i 101 = Ex5: i 12 i 25 i -3 =

51. For 2 complex numbers to be equal, the real parts must be equal and the imaginary parts must be equal. Ex1: 3x + 2i = 6 + 8yi Ex2: 4x – 3 + 2i = 9 – 6yi Ex3: 5 – (4 + y)i = 2x + 3 – 6i Assign: 1-24 all Can you define imaginary and complex numbers and perform simple operations on each?

52. 5.7a Solutions 2.7i20.x = -.5 y = -5/3 4.-9i22.x = 11/4 y = -2 6.24.x = 2/5 y = -4 8. 10.-i 12.i 14.-1 16.x = 1 y = -4 18.x = 2/3 y = -.5

53. 5.7b Operations on Complex Numbers Objective: To perform operations on complex numbers Add/subtract: Compare to: (4 – 3x) + (2x – 8) = Add: (2 + 4i) + (6 – 9i) = Subtract: (5 – 3i) – (7 – 5i) = -x - 4 Add real part to real part and imaginary part to imaginary part Multiply: Distribute or FOIL – all answers should be in standard complex form: a + bi Don’t forget i 2 = -1 Ex1: 2i(3 – 4i) =Ex2: -4i(5 + 6i) = Ex3: (2 – 3i)(4 + 5i) =Ex4: (4 – 2i) 2 =

54. Division: This is similar to rationalizing the denominator with radicals. Type 1: -or- Type 2: -or- Recall: How did we rationalize the denominator? Use the complex conjugate to divide complex numbers. a + bi a - bi Assign: 5.7b: 25-77 odd 87-90 Can you perform operations on complex numbers?

55. 5.7 b Solutions 88.i if n is even 90.x = 1 – i is a solution to the equation.

56. 5.7c and Review FOIL: (x - 3)(x + 3) -compare to- (x – 3i)(x + 3i) This means the following can be factored. How? 1.4x 2 - 25 2. 4x 2 + 25 3. x 2 + 44. 2x 2 + 98 Objective: To factor and simplify using complex numbers

57. Just for fun. (And they make great essay questions.) * What are imaginary numbers? * What symbol is used to designate imaginary units? * What is the value of the imaginary unit? * Give the definition of a complex number? * What is the complex conjugate and when should it be used? Give an example. * What is a pure imaginary number? Ex4: 4x – 3 + 2yi = x + 2y – 8i Ex5:

58. Can you factor and simplify using complex numbers? Assign Worksheet: Mini-Quiz Tomorrow!!

59. Worksheet Answers 1.10. 17i19. 28. 0 2. 10i11. 20. 29. 1 3.12. 621. 17-6i30. 1 4. -2013. -522. 5031. x 2 - 36 5. -814. 23. 32. x 2 + 36 6. i15. -8 + 6i24. -3 + 4i33. x 2 - 4 7. 15i16. 2/325. 1 + 21i34. x 2 + 4 8. -3517. -4i26. -45 + 30i35. (x+3)(x-3) 9. -3618. 27. 136. (x+3i)(x-3i) 37. (x+7)(x-7) 38. (x+7i)(x-7i)

60. Ch 5 Review Answers Ø

61. a b c a b a b