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1. 1. Exponents and Radicals Objective: To review rules and properties of exponents and radicals.
2. 2. Exponential Notation
3. 3. Properties of Exponents
4. 4. Properties of Exponents
5. 5. Example 1 • Use the properties of exponents to simplify each expression. a) ( 3ab 4 )( 4ab 3 )
6. 6. Example 1 • Use the properties of exponents to simplify each expression. a) ( 3ab 4 )( 4ab 3 ) ( 3ab 4 )( 4ab 3 ) 12 a 2b
7. 7. Example 1 • Use the properties of exponents to simplify each expression. You Try: b) 2 3 (2 xy )
8. 8. Example 1 • Use the properties of exponents to simplify each expression. You Try: b) 2 3 (2 xy ) (2 xy 2 )3 23 x 3 ( y 2 ) 3 8x3 y 6
9. 9. Example 1 • Use the properties of exponents to simplify each expression. You Try: c) 3a ( 4a 2 ) 0
10. 10. Example 1 • Use the properties of exponents to simplify each expression. You Try: c) 3a ( 4a 2 ) 0 3a( 4a 2 ) 0 3a(1) 3a, a 0
11. 11. Example 1 • Use the properties of exponents to simplify each expression. You Try: d) 5x y 3 2
12. 12. Example 1 • Use the properties of exponents to simplify each expression. You Try: d) 5x y 3 2 5x y 3 2 52 ( x 3 ) 2 y2 25x 6 y2
13. 13. Example 2 • Rewrite each expression with positive exponents. a) x 1
14. 14. Example 2 • Rewrite each expression with positive exponents. a) x 1 x 1 1 x
15. 15. Example 2 • Rewrite each expression with positive exponents. 1 b) 3x 2
16. 16. Example 2 • Rewrite each expression with positive exponents. 1 b) 3x 2 1 3x 2 1 1 3 x 2 x2 3
17. 17. Example 2 • Rewrite each expression with positive exponents. • You Try: 12 a 3b 4 c) 4a 2 b
18. 18. Example 2 • Rewrite each expression with positive exponents. • You Try: 12 a 3b 4 c) 4a 2 b 12 a 3b 4 4a 2 b 12 a 3 a 2 4b 4b 3a 5 b5
19. 19. Example 2 • Rewrite each expression with positive exponents. • You Try: d) 3x y 2 2
20. 20. Example 2 • Rewrite each expression with positive exponents. • You Try: d) 3x y 3x y 2 2 2 2 y 3x 2 2 y2 32 ( x 2 ) 2 y2 9x4
21. 21. Radicals and Their Properties • Definition of nth Root of a Number. • Let a and b be real numbers and let n > 2 be a positive integer. If a = bn then b is an nth root of a. If n = 2, the root is a square root. If n = 3, the root is a cube root.
22. 22. Radicals and Their Properties • Principal nth Root of a Number. • Let a be a real number that has at least one nth root. The principal nth root of a is the nth root that has the same sign as a. It is denoted by a radical symbol n a • The positive integer n is the index of the radical, and the number a is the radicand. If n = 2, omit the index and write a .
23. 23. Example 5 • Evaluate: a) 36
24. 24. Example 5 • Evaluate: a) 36 36 6
25. 25. Example 5 • Evaluate: b) 36
26. 26. Example 5 • Evaluate: b) 36 36 6
27. 27. Example 5 • Evaluate: c) 36
28. 28. Example 5 • Evaluate: c) 36 36 DNE
29. 29. Example 5 • Evaluate: d) 3 125 64
30. 30. Example 5 • Evaluate: d) 3 125 64 3 125 64 3 125 3 64 5 4
31. 31. Example 5 • Evaluate: • You Try: d) 3 27 8
32. 32. Example 5 • Evaluate: • You Try: d) 3 27 8 3 27 8 3 27 3 8 3 2
33. 33. Example 5 • Evaluate: e) 5 32
34. 34. Example 5 • Evaluate: e) 5 32 5 32 2
37. 37. Example 6 • Use the properties of radical to simplify each expression. a) 8 2
38. 38. Example 6 • Use the properties of radical to simplify each expression. a) 8 8 2 2 16 4
39. 39. Example 6 • Use the properties of radical to simplify each expression. b) 3 5 3
40. 40. Example 6 • Use the properties of radical to simplify each expression. b) 3 3 5 5 3 3 1/ 3 3 5 51 5
41. 41. Example 6 • Use the properties of radical to simplify each expression. c) 3 x 3
42. 42. Example 6 • Use the properties of radical to simplify each expression. c) 3 x x 3 3 1/ 3 x
43. 43. Example 6 • Use the properties of radical to simplify each expression. d) 6 6 y6 y 6 | y|
44. 44. Simplifying Radicals • An expression involving radicals is in simplest form when the following conditions are satisfied. 1. All possible factors have been removed from the radical. 2. All fractions have radical-free denominators (accomplished by a process called rationalizing the denominator). 3. The index of the radical is reduced.
45. 45. Example 7 • Simplify each radical. a) 32
46. 46. Example 7 • Simplify each radical. a) 32 32 16 2 4 2
47. 47. Example 7 • Simplify each radical. • You Try: a) 24
48. 48. Example 7 • Simplify each radical. • You Try: a) 24 24 4 6 2 6
49. 49. Example 7 • Simplify each radical. b) 4 48
50. 50. Example 7 • Simplify each radical. b) 4 4 48 48 4 16 4 3 24 3
51. 51. Example 7 • Simplify each radical. c) 75x3
52. 52. Example 7 • Simplify each radical. c) 75x3 75x3 25 3 x2 x 5x 3x
53. 53. Example 7 • Simplify each radical. • You Try: c) 48x5
54. 54. Example 7 • Simplify each radical. • You Try: c) 48x5 48x5 16 3 x4 x 4x2 3x
55. 55. Example 8 • Simplify each radical. a) 3 24
56. 56. Example 8 • Simplify each radical. a) 3 24 3 24 3 8 3 3 23 3
57. 57. Example 8 • Simplify each radical. b) 3 24a 4
58. 58. Example 8 • Simplify each radical. b) 3 24a 4 3 4 24a 3 8 3 3 3 a 3 3 a 2a3 3a
59. 59. Example 8 • Simplify each radical. • You Try: c) 3 40x6
60. 60. Example 8 • Simplify each radical. • You Try: c) 3 40x6 3 40x6 3 8 3 5 3 x6 2x 2 3 5
61. 61. Example 9 • Combine each radical. a) 2 48 3 27
62. 62. Example 9 • Combine each radical. a) 2 48 3 27 2 48 3 27 2 16 3 3 8 9 3 9 3 3 3
63. 63. Example 9 • Combine each radical. • You Try: b) 3 16x 3 54x 4
64. 64. Example 9 • Combine each radical. • You Try: b) 3 3 16x 3 16x 3 54x 4 54x 4 3 8 2 3 3 2x 3 27x 2 x 3x 3 3 2x 3 2x ( 2 3 x )3 2 x
65. 65. Example 10 • Rationalize the denominator of each expression. a) 5 3
66. 66. Example 10 • Rationalize the denominator of each expression. a) 5 3 5 3 3 3 5 3 3
67. 67. You Try • Rationalize the denominator of each expression. • You Try: b) 1 2
68. 68. Example 10 • Rationalize the denominator of each expression. • You Try: b) 1 2 1 2 2 2 2 2
69. 69. Example 11 • Rationalize the denominator of each expression. 2 a) 3 7 2 (3 (3 7 ) (3 7) 7) 2(3 7 ) 9 7 3 7
70. 70. You Try • Rationalize the denominator of each expression. • You Try: b) 3 4 5
71. 71. You Try • Rationalize the denominator of each expression. • You Try: b) 3 4 5 3 (4 (4 5 ) (4 5) 5) 12 3 5 11
72. 72. Rational Exponents • The numerator of a rational exponent denotes the power to which the base is raised, and the denominator denotes the index or the root to be taken.
73. 73. Example 13 • Change the base from radical to exponential form. a) 3
74. 74. Example 13 • Change the base from radical to exponential form. a) 3 3 1/ 2 3
75. 75. Example 13 • Change the base from radical to exponential form. b) x 5
76. 76. Example 13 • Change the base from radical to exponential form. b) x 5 x 5 5 1/ 2 (x ) x 5/ 2
77. 77. You Try • Change the base from radical to exponential form. • You Try: c) 3 y 4
78. 78. You Try • Change the base from radical to exponential form. • You Try: c) 3 3 y 4 y 4 4 1/ 3 (y ) y 4/3
79. 79. Example 14 • Change the base from exponential to radical form. a) ( x y ) 3 / 2
80. 80. Example 14 • Change the base from exponential to radical form. a) ( x y ) 3 / 2 ( x y)3 / 2 ( x y)3
81. 81. You Try • Change the base from exponential to radical form. b) x 3 / 4 y1/ 4
82. 82. You Try • Change the base from exponential to radical form. b) x 3 / 4 y1/ 4 x3 / 4 y1/ 4 ( x3 y)1/ 4 4 x3 y
83. 83. Example 15 • Simplify each rational expression. a) ( 32 ) 3 / 5
84. 84. Example 15 • Simplify each rational expression. a) ( 32 ) 3 / 5 ( 32 ) 3/ 5 1/ 5 3 (( 32 ) ) ( 2) 3 8
85. 85. Example 15 • Simplify each rational expression. • You Try: b) (27 ) 2/3
86. 86. Example 15 • Simplify each rational expression. • You Try: b) (27 ) 2/3 (27) 2/3 1/ 3 ((27) ) 2 (3) 2 1 9
87. 87. You Try • Simplify each rational expression. • You Try: c) (64 ) 2 / 3
88. 88. You Try • Simplify each rational expression. • You Try: c) (64 ) 2 / 3 (64 ) 2/3 1/ 3 2 (( 64 ) ) 4 2 16
89. 89. You Try • Simplify each rational expression. • You Try: d) (16 ) 3/ 4
90. 90. You Try • Simplify each rational expression. • You Try: d) (16 ) (16) 3/ 4 3/ 4 1/ 4 ((16) ) 3 2 3 1 8