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1 Roots & Radicals Intermediate Algebra

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

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3 Radicals 7.1

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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)

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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,

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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.

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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.

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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).

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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)

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

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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

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12 Radicals 7-8Page 397

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13 Square Root of x 2 7-7Page 393

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14 Product Rule for Radicals 7-9Page 398

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15 Simplifying Radical Expressions Product Rule –

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16 Quotient Rule for Radicals 7-10Page 399

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17 Quotient Rule for Radicals 7-10Page 399

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18 Quotient Rule for Radicals 7-10Page 399

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

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

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21 Warm-Ups 7.1

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

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23 Wind Chill

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24 Wind Chill

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25 Wind Chill

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26 Wind Chill

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27 Rational Exponents 7.2

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

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29 When n Is Even

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

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31 Exponent 1/n When n Is Odd

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32 nth Root of Zero Page 389

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33 Rational Exponents 7-4Page 390

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34 Evaluating in Either Order

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35 Negative Rational Exponents 7-5Page 391

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36 Evaluating a - m/n

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37 Rules for Rational Exponents 7-6Page 392

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38 Simplifying

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39 Simplifying

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40 Simplifying

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41 Simplifying

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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.

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43 Warm-Ups 7.2

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

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45 California Growing

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46 Growth Rate

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47 Operations with Radicals 7.3

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48 Addition and Subtraction Like Radicals

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49 Addition and Subtraction Like Radicals

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50 Simplifying Before Combining

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51 Simplifying Before Combining

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52 Simplifying Before Combining

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53 Simplifying Before Combining

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54 Simplifying Before Combining

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55 Simplifying Before Combining

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56 Simplifying Before Combining

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57 Simplifying Before Combining

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58 Simplifying Before Combining

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59 Simplifying Before Combining

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60 Multiplying Radicals Same index

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61 Multiplying Radicals Same index

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62 Multiplying Radicals Same index

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63 Multiplying Radicals Same index

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64 Multiplying Radicals Same index

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65 Multiplying Radicals Same index

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66 Multiplying Radicals Same index

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67 Multiplying Radicals Same index

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68 Multiplying Radicals - Binomials

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69 Multiplying Binomials

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70 Multiplying Binomials

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71 Multiplying Binomials

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72 Multiplying Binomials

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73 Multiplying Radicals – Different Indices

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74 Multiplying Radicals Different Indices

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75 Different Indices

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76 Different Indices

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77 Different Indices

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78 Conjugates

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79 Conjugates

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80 Warm-Ups 7.3

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

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82 Area of a Triangle

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83 Area of a Triangle

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84 Area of a Triangle

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85 Quotients, Powers, etc 7.4

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86 Dividing Radicals

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87 Dividing Radicals

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88 Dividing Radicals

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89 Rationalizing the Denominator

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90 Rationalizing the Denominator

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91 Rationalizing the Denominator

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92 Rationalizing the Denominator

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93 Powers of Radical Expressions

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94 Powers of Radical Expressions

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95 Warm-Ups 7.4

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

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97 7.4 #102

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98 Adding Fractions

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99 Adding Fractions

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100 Solving Equations 7.5

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

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

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

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

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105 Even-Root Property 7-11Page 419

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106 Even-Root Property 7-11Page 419

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

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

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

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

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111 Isolating the Radical

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112 Squaring Both Sides

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113 Cubing Both Sides

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114 Squaring Both Sides Twice

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115 Squaring Both Sides Twice

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116 Squaring Both Sides Twice

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117 Rational Exponents Eliminate the root, then the power

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118 Eliminate the Root, Then the Power

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119 Negative Exponents

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120 Negative Exponents Eliminate the root, then the power

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121 Negative Exponents Eliminate the root, then the power

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122 No Solution Eliminate the root, then the power

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123 No Solution Eliminate the root, then the power

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

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

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

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

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

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

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

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131 Warm-Ups 7.5

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

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133 Complex Numbers 7.6

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134 Imaginary Numbers

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135 Imaginary Numbers

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136 Imaginary Numbers

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137 Imaginary Numbers

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138 Imaginary Numbers

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139 Imaginary Numbers

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140 Powers of i

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141 Complex Numbers 7-14Page 429

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

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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

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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

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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

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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

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

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

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

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

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

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

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153 Imaginary Solutions to Equations

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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

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155 Complex Numbers

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156 Warm-Ups 7.6

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

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