1. 1/8/2016Math 120 - KM1 Chapter 8: Radical Expressions, Equations, and Functions 8.1 Radical Expressions & Functions 8.2 Rational Numbers as Exponents 8.3 Simplifying Radical Expressions 8.4 Addition, Subtraction, and more Multiplication 8.5 More on Division of Radical Expressions 8.6 Solving Radical Equations 8.7 Applications involving Powers and Roots 8.8 The Complex Numbers

2. 1/8/2016Math 120 - KM2 8.1 Radical Expressions & Functions

3. 1/8/2016Math 120 - KM3 Index The “default” index is 2. Radicand Parts of a Radical means 8.1

4. 1/8/2016Math 120 - KM4 What’s the difference? What are the square root(s) of 49? The square root(s) of 49 are -7 and 7. Every number (except zero) has two square roots. 8.1

5. 1/8/2016Math 120 - KM5 Is the radical different? Well then, what is ? is the principal square root of 49. 8.1

6. 1/8/2016Math 120 - KM6 Counting with Radicals 8.1

7. 1/8/2016Math 120 - KM7 Simplify a few? Not a Real Number 8.1

8. 1/8/2016Math 120 - KM8 Exact vs Approximate is an EXACT value. is an APPROXIMATE value. 8.1

9. 1/8/2016Math 120 - KM9 The Square Root Function The DOMAIN is x > 0 xf(x) 00 11 42 93 8.1

10. 1/8/2016Math 120 - KM10 A minor change? The DOMAIN is x > 2 xf(x) 20 31 62 113 8.1

11. 1/8/2016Math 120 - KM11 Random Radicals ? 8.1

12. 1/8/2016Math 120 - KM12 Tricky Problems? Absolutely! 6.1

13. 1/8/2016Math 120 - KM13 6.2 Rational Numbers as Exponents 8.2

14. 1/8/2016Math 120 - KM14 The Basic Idea! 8.2

15. 1/8/2016Math 120 - KM15 See How this Works? 8.2

16. 1/8/2016Math 120 - KM16 Let’s try another! 8.2

17. 1/8/2016Math 120 - KM17 Isn’t this fun? 8.2

18. 1/8/2016Math 120 - KM18 Negative? OK! 6.2

19. 1/8/2016Math 120 - KM19 Try this one? 8.2

20. 1/8/2016Math 120 - KM20 Where We Left Off Last Class

21. 1/8/2016Math 120 - KM21 Simplify? 8.2

22. 1/8/2016Math 120 - KM22 Play by the Rules! 8.2

23. 1/8/2016Math 120 - KM23 What if? 8.2

24. 1/8/2016Math 120 - KM24 You can do this! 8.2

25. 1/8/2016Math 120 - KM25 Rewrite in radical form Reverse the Process? 8.2

26. 1/8/2016Math 120 - KM26 Rewrite in exponential form OK...now the other way! 8.2

27. 1/8/2016Math 120 - KM27 Conquer This! 8.2

28. 1/8/2016Math 120 - KM28 8.3 Simplifying Radical Expressions 8.3

29. 1/8/2016Math 120 - KM29 Simplify: Assume that all expressions under radicals represent nonnegative numbers. 25.2.2 8.3

30. 1/8/2016Math 120 - KM30 Simplify: -27 8.3

31. 1/8/2016Math 120 - KM31 Simplify: -32.3.3 8.3

32. 1/8/2016Math 120 - KM32 -8.7.7 8.3

33. 1/8/2016Math 120 - KM33 4.5.5 8.3

34. 1/8/2016Math 120 - KM34 Product Property are Real numbers 25.2.2 8.3

35. 1/8/2016Math 120 - KM35 8.5.5 8.3

36. 1/8/2016Math 120 - KM36 8.3

37. 1/8/2016Math 120 - KM37 Quotient Property are Real numbers 8.3

38. 1/8/2016Math 120 - KM38 8.3

39. 1/8/2016Math 120 - KM39 6.4 Addition, Subtraction, and more Multiplication 8.4

40. 1/8/2016Math 120 - KM40 16.2.2 25.2.2 8.4

41. 1/8/2016Math 120 - KM41 27.3.3 64.3.3 8.4

42. 1/8/2016Math 120 - KM42 8.4

43. 1/8/2016Math 120 - KM43 4.3.3 8.4

44. 1/8/2016Math 120 - KM44 8.4

45. 1/8/2016Math 120 - KM45 8.4

46. 1/8/2016Math 120 - KM46 8.4

47. 1/8/2016Math 120 - KM47 8.5 More on Division of Radical Expressions 8.5

48. 1/8/2016Math 120 - KM48 8.5

49. 1/8/2016Math 120 - KM49 8.5

50. 1/8/2016Math 120 - KM50 8.5

51. 1/8/2016Math 120 - KM51 8.5

52. 1/8/2016Math 120 - KM52 8.5

53. 1/8/2016Math 120 - KM53 8.6 Solving Radical Equations 8.6

54. 1/8/2016Math 120 - KM54 ISOLATE the RADICAL Raise to the power of the index Check for extraneous solutions More than one radical?… separate the radicals to opposite sides of the equation and power up!! 8.6

55. 1/8/2016Math 120 - KM55 An equation that contains a variable expression in a radicand is a radical equation. A POWER RULE for EQUATIONS 8.6

56. 1/8/2016Math 120 - KM56 8.6

57. 1/8/2016Math 120 - KM57 8.6

58. 1/8/2016Math 120 - KM58 8.6

59. 1/8/2016Math 120 - KM59 8.6

60. 1/8/2016Math 120 - KM60 8.6

61. 1/8/2016Math 120 - KM61 8.6

62. 1/8/2016Math 120 - KM62 Be sure to check your solutions! 8.6

63. 1/8/2016Math 120 - KM63 8.7 Applications involving Powers and Roots 8.7

64. 1/8/2016Math 120 - KM64 An object is dropped from a bridge. Find the distance the object has fallen when its speed reaches 120 ft/s. Use the equation, where v is the speed of the object in feet per second an d is the distance in feet. 8.7

65. 1/8/2016Math 120 - KM65 An 18 foot ladder is leaning against a building. How high on the building will the ladder reach when the base of the ladder is 6 feet from the building? 8.7

66. 1/8/2016Math 120 - KM66 8.8 The Complex Numbers 8.8

67. 1/8/2016Math 120 - KM67 The original i... The term “imaginary number” was coined in 1637 by Rene Descartes Several subjects in physics require complex numbers, such as quantum mechanics, general relativity and fluid dynamics. Also, complex numbers play a key role in chaos theory and in fractal geometry. http://mathchaostheory.suite101.com/article.cfm/complex_numbers Imaginary numbers were defined in 1572 by Rafael Bombelli an Italian mathematician.Rafael Bombelli 8.8

68. 1/8/2016Math 120 - KM68 More uses for i... Complex numbers are used extensively in physics to describe Electromagnetic Waves and Quantum Mechanics. http://www.jamesbrennan.org/jbrennan/139/notes/Complex%20Numbers/complex_numbers.htm In electrical engineering complex numbers are used to represent the phase of an alternating signal affected by inductance and capacitance. However, the actual voltage or current at any time is still a real number (which is calculated from the complex number). 8.8 http://www.articlesbase.com/k-12-education-articles/mathematics-in-physics-and-chemistry-893862.html

69. 1/8/2016Math 120 - KM69 Aerodynamics too… The mapping function gives the velocity and pressures around the airfoil. Knowing the pressure around the airfoil, allows the “lift” to be determined. 6.8 http://www.grc.nasa.gov/WWW/K-12/airplane/map.html

70. 1/8/2016Math 120 - KM70 Imaginary... Not really! http://en.wikipedia.org/wiki/Mandelbrot_set 8.8 “Mathematically, the Mandelbrot set can be defined as the set of complex c-values for which the orbit of 0 under iteration of the complex quadratic polynomial x n+1 =x n 2 + c remains bounded.” http://plus.maths.org/issue40/features/devaney/

71. 1/8/2016Math 120 - KM71 o The Complex Plane / Unit Circle Imaginary axis Real axis 8.8

72. 1/8/2016Math 120 - KM72 Imaginary axis Real axis Powers of “i” ? Use the i-clock 8.8

73. 1/8/2016Math 120 - KM73 Use the i-clock 8.8

74. 1/8/2016Math 120 - KM74 Use your i-magination! 8.8

75. 1/8/2016Math 120 - KM75 Try this one? 100.5.5 8.8

76. 1/8/2016Math 120 - KM76 Complex it is! 8.8

77. 1/8/2016Math 120 - KM77 But not too complex for you! 25.2.2 8.8

78. 1/8/2016Math 120 - KM78 ( 11 + 4i ) + ( 9 + 3i ) i bet you can do this! = 20 + 7i 8.8

79. 1/8/2016Math 120 - KM79 ( -4 + 2i ) – ( 7 – 3i ) No problem...right? = -11 + 5i 8.8

80. 1/8/2016Math 120 - KM80 ( 3 + 2i ) + ( 3 – 2i ) Add Complex Conjugates? Really? = 6 These numbers add to 6 and multiply to 13 8.8

81. 1/8/2016Math 120 - KM81 Rewrite, then Simplify 4.3.3 36.3.3 8.8

82. 1/8/2016Math 120 - KM82 (3i)(5i) Multiply? i remember! = 15i 2 = -15 8.8

83. 1/8/2016Math 120 - KM83 (-8i)(7i) Here’s another! = -56i 2 = 56 8.8

84. 1/8/2016Math 120 - KM84 (-6i)(-2i) One More? = 12i 2 = -12 8.8

85. 1/8/2016Math 120 - KM85 ( 4 + 3i)(5 – i) FOiL... i know you can! = 20 – 4i + 15i – 3i 2 = 23 + 11i = 20 – 4i + 15i + 3 8.8

86. 1/8/2016Math 120 - KM86 (4 + 3i)(4 – 3i) Product of Complex Conjugates = 16 – 9i 2 = 16 + 9 = 25 These numbers multiply to 25 and add to 8 8.8

87. 1/8/2016Math 120 - KM87 Convert to i then Distribute 8.8

88. 1/8/2016Math 120 - KM88 “Real-ize” the denominator! 8.8

89. 1/8/2016Math 120 - KM89 Another Denominator to “Realize”. 8.8

90. 1/8/2016Math 120 - KM90 Conjugate Time! 8.8

91. 1/8/2016Math 120 - KM91 Another Reality Check? 8.8

92. 1/8/2016Math 120 - KM92 Solution Check? Is 1 + 2i a solution of x 2 – 2x + 5 = 0 ? 8.8

93. 1/8/2016Math 120 - KM93 Seattle Fractals http://www.fractalarts.com/ASF/NEW.html Amazing Seattle Fractals! Fractal Art, Screensavers, Tutorials, Software & more! Doug Harrington

94. 1/8/2016Math 120 - KM94 That’s All For Now!