Download presentation

Presentation is loading. Please wait.

Published byVirgil Simon Barton Modified over 5 years ago

1
1 Roots & Radicals Intermediate Algebra

2
2 Roots and Radicals Radicals Rational Exponents Operations with Radicals Quotients, Powers, etc. Solving Equations Complex Numbers

3
3 Radicals 7.1

4
4 Square Roots Finding Square Roots 3 2 = 9 (-3) 2 = 9N.B. -3 2 = -9 (½) 2 = (¼) The square root of 9 is 3 The square root of 9 is also –3 The square root of (¼) is (½)

5
5 Square Roots The square root symbol Radical sign The expression within is the radicand Square Root If a is a positive number, then is the positive square root of a is the negative square root of a Also,

6
6 Approximating Square Roots Perfect squares are numbers whose square roots are integers, for example 81 = 9 2. Square roots of other numbers are irrational numbers, for example We can approximate square roots with a calculator.

7
7 Approximating Square Roots 3.162(Calculator) We can determine that it is greater than 3 and less then 4 because 3 2 = 9 and 4 2 =16.

8
8 Cube Roots 2 is the cube root of 8 because 2 3 = 8. 8 and 2 3 above are radicands 3 is called the index (index 2 is omitted).

9
9 Cube Roots Evaluated 2 is the cube root of 8 because 2 3 = 8. 8 and 2 3 above are radicands 3 is called the index (index 2 is omitted)

10
10 nth Roots The number b is an nth root of a,, if b n = a.

11
11 nth Roots An nth root of number a is a number whose nth power is a. a number whose nth power is a If the index n is even, then the radicand a must be nonnegative. is not a real number

12
12 Radicals 7-8Page 397

13
13 Square Root of x 2 7-7Page 393

14
14 Product Rule for Radicals 7-9Page 398

15
15 Simplifying Radical Expressions Product Rule –

16
16 Quotient Rule for Radicals 7-10Page 399

17
17 Quotient Rule for Radicals 7-10Page 399

18
18 Quotient Rule for Radicals 7-10Page 399

19
19 Radical Functions Finding the domain of a square root function.

20
20 Radical Functions Finding the domain of a square root function.

21
21 Warm-Ups 7.1

22
22 7.1 T or F 1.T6. F 2.F7. F 3.T8. F 4.F9. T 5.T10. T

23
23 Wind Chill

24
24 Wind Chill

25
25 Wind Chill

26
26 Wind Chill

27
27 Rational Exponents 7.2

28
28 Exponent 1/n When n Is Even 7-1Page 388

29
29 When n Is Even

30
30 Exponent 1/n When n Is Odd 7-2Page 389

31
31 Exponent 1/n When n Is Odd

32
32 nth Root of Zero Page 389

33
33 Rational Exponents 7-4Page 390

34
34 Evaluating in Either Order

35
35 Negative Rational Exponents 7-5Page 391

36
36 Evaluating a - m/n

37
37 Rules for Rational Exponents 7-6Page 392

38
38 Simplifying

39
39 Simplifying

40
40 Simplifying

41
41 Simplifying

42
42 Simplified Form for Radicals of Index n A radical expression of index n is in Simplified Radical Form if it has 1.No perfect nth powers as factors of the radicand, 2.No fractions inside the radical, and 3.No radicals in the denominator.

43
43 Warm-Ups 7.2

44
44 7.2 T or F 1.T6. T 2.F7. T 3.F8. F 4.T9. T 5.T10. T

45
45 California Growing

46
46 Growth Rate

47
47 Operations with Radicals 7.3

48
48 Addition and Subtraction Like Radicals

49
49 Addition and Subtraction Like Radicals

50
50 Simplifying Before Combining

51
51 Simplifying Before Combining

52
52 Simplifying Before Combining

53
53 Simplifying Before Combining

54
54 Simplifying Before Combining

55
55 Simplifying Before Combining

56
56 Simplifying Before Combining

57
57 Simplifying Before Combining

58
58 Simplifying Before Combining

59
59 Simplifying Before Combining

60
60 Multiplying Radicals Same index

61
61 Multiplying Radicals Same index

62
62 Multiplying Radicals Same index

63
63 Multiplying Radicals Same index

64
64 Multiplying Radicals Same index

65
65 Multiplying Radicals Same index

66
66 Multiplying Radicals Same index

67
67 Multiplying Radicals Same index

68
68 Multiplying Radicals - Binomials

69
69 Multiplying Binomials

70
70 Multiplying Binomials

71
71 Multiplying Binomials

72
72 Multiplying Binomials

73
73 Multiplying Radicals – Different Indices

74
74 Multiplying Radicals Different Indices

75
75 Different Indices

76
76 Different Indices

77
77 Different Indices

78
78 Conjugates

79
79 Conjugates

80
80 Warm-Ups 7.3

81
81 7.3 T or F 1.F6. F 2.T7. T 3.F8. F 4.F9. F 5.T10. T

82
82 Area of a Triangle

83
83 Area of a Triangle

84
84 Area of a Triangle

85
85 Quotients, Powers, etc 7.4

86
86 Dividing Radicals

87
87 Dividing Radicals

88
88 Dividing Radicals

89
89 Rationalizing the Denominator

90
90 Rationalizing the Denominator

91
91 Rationalizing the Denominator

92
92 Rationalizing the Denominator

93
93 Powers of Radical Expressions

94
94 Powers of Radical Expressions

95
95 Warm-Ups 7.4

96
96 7.4 T or F 1.T6. T 2.T7. F 3.F8. T 4.T9. T 5.F10. T

97
97 7.4 #102

98
98 Adding Fractions

99
99 Adding Fractions

100
100 Solving Equations 7.5

101
101 Solving Equations The Odd Root Property If n is an odd positive integer, for any real number k.

102
102 Solving Equations – Odd Powers The Odd Root Property If n is an odd positive integer, for any real number k.

103
103 Solving Equations – Odd Powers The Odd Root Property If n is an odd positive integer, for any real number k.

104
104 Solving Equations – Odd Powers The Odd Root Property If n is an odd positive integer, for any real number k.

105
105 Even-Root Property 7-11Page 419

106
106 Even-Root Property 7-11Page 419

107
107 Solving Equations – Even Powers The Even Root Property If n is an even positive integer,

108
108 Solving Equations – Even Powers The Even Root Property If n is an even positive integer,

109
109 Solving Equations – Even Powers The Even Root Property If n is an even positive integer,

110
110 Solving Equations – Even Powers The Even Root Property If n is an even positive integer,

111
111 Isolating the Radical

112
112 Squaring Both Sides

113
113 Cubing Both Sides

114
114 Squaring Both Sides Twice

115
115 Squaring Both Sides Twice

116
116 Squaring Both Sides Twice

117
117 Rational Exponents Eliminate the root, then the power

118
118 Eliminate the Root, Then the Power

119
119 Negative Exponents

120
120 Negative Exponents Eliminate the root, then the power

121
121 Negative Exponents Eliminate the root, then the power

122
122 No Solution Eliminate the root, then the power

123
123 No Solution Eliminate the root, then the power

124
124 Strategy for Solving Equations with Exponents and Radicals 7-12Page 424

125
125 Distance Formula 7-13Page 424 Pythagorean Theorema 2 + b 2 = c 2

126
126 Distance Formula 7-13Page 424 Find the distance between the points (-2,3) and (1, -4).

127
127 Distance Formula 7-13Page 424 Find the distance between the points (-2,3) and (1,-4).

128
128 Diagonal of a Sign What is the length of the diagonal of a rectangular billboard whose sides are 5 meters and 12 meters?

129
129 Diagonal of a Sign What is the length of the diagonal of a rectangular billboard whose sides are 5 meters and 12 meters?

130
130 Diagonal of a Sign What is the length of the diagonal of a rectangular billboard whose sides are 5 meters and 12 meters?

131
131 Warm-Ups 7.5

132
132 7.5 T or F 1.F6. F 2.T7. F 3.F8. T 4.F9. T 5.T10. T

133
133 Complex Numbers 7.6

134
134 Imaginary Numbers

135
135 Imaginary Numbers

136
136 Imaginary Numbers

137
137 Imaginary Numbers

138
138 Imaginary Numbers

139
139 Imaginary Numbers

140
140 Powers of i

141
141 Complex Numbers 7-14Page 429

142
142 Figure 7.3 7-15Page 430 (Figure 7.3)

143
143 Addition and Subtraction The sum and difference a + bi of c + di and are: (a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) - (c + di) = (a - c) + (b - d)i

144
144 (2 + 3i) + (4 + 5i) The sum and difference a + bi of c + di and are: (2 + 3i) + (4 + 5i) = (2 + 4) + (3 + 5)i = 6 + 8i (2 + 3i) – (4 + 5i) = (2 – 4) + (3 – 5)i = – 2 – 2i

145
145 Multiplication The complex numbers a + bi of c + di and are multiplied as follows: (a + bi) (c + di) = ac + adi + bci + bdi 2 = ac + bd(– 1) + adi + bci = (ac – bd) + (ad + bc)i

146
146 (2 + 3i) (4 + 5i) The complex numbers a + bi of c + di and are multiplied as follows: (a + bi) (c + di) = (ac – bd) + (ad + bc)i (2 + 3i) (4 + 5i) = 8 + 10i + 12i + 15i 2 = 8 + 22i + 15(– 1) = – 7 + 22i

147
147 Division (2 + 3i) ÷ 4 = (2 + 3i) / 4 = ½ + ¾ i

148
148 Complex Conjugates The complex numbers a + bi and a – bi are called complex conjugates. Their product is a 2 + b 2.

149
149 Division We divide the complex number a + bi by the complex number c + di as follows:

150
150 Division We divide the complex number a + bi by the complex number c + di as follows:

151
151 Division We divide the complex number 2 + 3i by the complex number 4 + 5i.

152
152 Square Root of a Negative Number For any positive real number b,

153
153 Imaginary Solutions to Equations

154
154 Complex Numbers 1.Definition of i: i =, i 2 = -1. 2.Complex number form: a + bi. 3. a + 0i is the real number a. 4. b is a positive real number 5.The numbers a + bi and a - bi are complex conjugates. Their product is a 2 + b 2. 6.Add, subtract, and multiply complex numbers as if they were algebraic expressions with i being the variable, and replace i 2 by -1. 7.Divide complex numbers by multiplying the numerator and denominator by the conjugate of the denominator. 8.In the complex number system x 2 = k for any real number k is equivalent to

155
155 Complex Numbers

156
156 Warm-Ups 7.6

157
157 7.6 T or F 1.T6. T 2.F7. T 3.F8. F 4.T9. T 5.T10. F

158
158

159
159

Similar presentations

© 2020 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google