## Presentation on theme: "1 Roots & Radicals Intermediate Algebra. 2 Roots and Radicals Radicals Rational Exponents Operations with Radicals Quotients, Powers, etc. Solving Equations."— Presentation transcript:

1 Roots & Radicals Intermediate Algebra

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

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

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

13 Square Root of x 2 7-7Page 393

14 Product Rule for Radicals 7-9Page 398

15 Simplifying Radical Expressions Product Rule –

16 Quotient Rule for Radicals 7-10Page 399

17 Quotient Rule for Radicals 7-10Page 399

18 Quotient Rule for Radicals 7-10Page 399

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

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

21 Warm-Ups 7.1

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

23 Wind Chill

24 Wind Chill

25 Wind Chill

26 Wind Chill

27 Rational Exponents 7.2

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

29 When n Is Even

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

31 Exponent 1/n When n Is Odd

32 nth Root of Zero Page 389

33 Rational Exponents 7-4Page 390

34 Evaluating in Either Order

35 Negative Rational Exponents 7-5Page 391

36 Evaluating a - m/n

37 Rules for Rational Exponents 7-6Page 392

38 Simplifying

39 Simplifying

40 Simplifying

41 Simplifying

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 Warm-Ups 7.2

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

45 California Growing

46 Growth Rate

50 Simplifying Before Combining

51 Simplifying Before Combining

52 Simplifying Before Combining

53 Simplifying Before Combining

54 Simplifying Before Combining

55 Simplifying Before Combining

56 Simplifying Before Combining

57 Simplifying Before Combining

58 Simplifying Before Combining

59 Simplifying Before Combining

69 Multiplying Binomials

70 Multiplying Binomials

71 Multiplying Binomials

72 Multiplying Binomials

73 Multiplying Radicals – Different Indices

75 Different Indices

76 Different Indices

77 Different Indices

78 Conjugates

79 Conjugates

80 Warm-Ups 7.3

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

82 Area of a Triangle

83 Area of a Triangle

84 Area of a Triangle

85 Quotients, Powers, etc 7.4

89 Rationalizing the Denominator

90 Rationalizing the Denominator

91 Rationalizing the Denominator

92 Rationalizing the Denominator

95 Warm-Ups 7.4

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

97 7.4 #102

100 Solving Equations 7.5

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

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

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

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

105 Even-Root Property 7-11Page 419

106 Even-Root Property 7-11Page 419

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

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

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

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

112 Squaring Both Sides

113 Cubing Both Sides

114 Squaring Both Sides Twice

115 Squaring Both Sides Twice

116 Squaring Both Sides Twice

117 Rational Exponents Eliminate the root, then the power

118 Eliminate the Root, Then the Power

119 Negative Exponents

120 Negative Exponents Eliminate the root, then the power

121 Negative Exponents Eliminate the root, then the power

122 No Solution Eliminate the root, then the power

123 No Solution Eliminate the root, then the power

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

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

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

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

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

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 Warm-Ups 7.5

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

133 Complex Numbers 7.6

134 Imaginary Numbers

135 Imaginary Numbers

136 Imaginary Numbers

137 Imaginary Numbers

138 Imaginary Numbers

139 Imaginary Numbers

140 Powers of i

141 Complex Numbers 7-14Page 429

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

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

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

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

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

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

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

153 Imaginary Solutions to Equations

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 Complex Numbers

156 Warm-Ups 7.6

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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