Slide 1 / 180. Radicals and Rational Exponents

Transcription

1 Slide 1 / 180 Radicals and Rational Exponents

2 Slide 2 / 180 Roots and Radicals Table of Contents: Square Roots Intro to Cube Roots n th Roots Irrational Roots Rational Exponents Operations with Radicals Addition and Subtraction Multiplication Division: Rationalizing the Denominator Complex Numbers

3 Slide 3 / 180 Roots

4 Slide 4 / 180 The symbol for taking a square root is, it is a radical sign. The square root cancels out the square. There is no real square root of a negative number. is not real (4 2 =16 and (-4) 2 =16)

5 1 What is 1? Slide 5 / 180

6 2 What is? Slide 6 / 180

7 3 What is? Slide 7 / 180

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9 Slide 9 / 180 To take the square root of a variable rewrite its exponent as the square of a power. Square roots need to be positive answers. Even powered answered, like above, are positive even if the variables negative. The same cannot be said if the answer has an odd power. When you take a square root an the answer has an odd power, put the answer inside of absolute value signs.

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16 Slide 16 / A C B D no real solution

17 Slide 17 / A C B D no real solution

18 Slide 18 / A C B D no real solution

19 Slide 19 / A C B D no real solution

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21 Slide 21 / Evaluate A B C D No Real Solution

22 Slide 22 / Evaluate A B C D No Real Solution

23 Slide 23 / Evaluate A B C D No Real Solution

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25 Slide 25 / 180 Intro to Cube Roots Return to Table of Contents

26 Slide 26 / 180 Q: If a square root cancels a square, what cancels a cube? A: A cube root.

27 Slide 27 / 180 The volume (V) of a cube is found by cubing its side length (s). V = s 3 V = s 3 V = 4 3 = V = 64 cubic units or 64 units 3 4 units The volume (V) of a cube is labeled as cubic units, or units 3, because to find the volume, you need to cube its side.

28 Slide 28 / 180 A cube with sides 3 units would have a volume of 27 u 3 because 3 3 =27. If a cube has an volume of 64 u 3 what is the length of one side? Need to find a number when multiplied by itself three times will equal = 64, so 4 units is the length of a side.

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39 Slide 39 / Simplify A B C D not possible

40 Slide 40 / Simplify A B C D not possible

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43 Slide 43 / Which of the following is not a step in simplifying A C B D

44 Slide 44 / 180 nth Roots Return to Table of Contents

45 Slide 45 / 180 In general, and absolute value signs are needed if n is even and the variable has an odd powered answer.

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48 Slide 48 / Simplify A B C D

49 Slide 49 / Simplify A B C D

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51 Slide 51 / Simplify A B C D

52 Slide 52 / Simplify A B C D

53 Slide 53 / Simplify A B C D

54 Slide 54 / If the n th root of a radicand is, which of the following is always true? A B C D No absolute value signs are ever needed. Absolute value signs will always be needed. Absolute value signs will be needed if j is negative. Absolute value signs are needed if n is an even index.

55 Slide 55 / 180 Rational Exponents Return to Table of Contents

56 Slide 56 / 180 Rational Exponents, or exponents that are fractions, is another way to write a radical.

57 Slide 57 / 180 Rewrite each radical as a rational exponent in the lowest terms.

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61 Slide 61 / Find the simplified expression that is equivalent to: A B C D

62 Slide 62 / Find the simplified expression that is equivalent to: A B C D

63 Slide 63 / Simplify A C B D

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65 Slide 65 / Simplify A B C D

66 Slide 66 / Simplify A B C D

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68 Slide 68 / Find the simplified expression that is equivalent to: A B C D

69 Slide 69 / Find the simplified expression that is equivalent to: A B C D

70 Slide 70 / 180 Simplifying Radicals is said to be a rational answer because their is a perfect square that equals the radicand. If a radicand doesn't have a perfect square that equals it, the root is said to be irrational.

71 Slide 71 / 180 The square root of the following numbers is rational or irrational?

72 Slide 72 / 180 The commonly excepted form of a radical is called the "simplified form To simplify a non-perfect square, start by breaking the radicand into factors and then breaking the factors into factors and so on until there only prime numbers are left. this is called the prime factorization.

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74 Slide 74 / Which of the following is the prime factorization of 24? A 3(8) B 4(6) C 2(2)(2)(3) D 2(2)(2)(3)(3)

75 Slide 75 / Which of the following is the prime factorization of 72? A 9(8) B 2(2)(2)(2)(6) C 2(2)(2)(3) D 2(2)(2)(3)(3)

76 Slide 76 / Which of the following is the prime factorization of 12? A 3(4) B 2(6) C 2(2)(2)(3) D 2(2)(3)

77 Slide 77 / Which of the following is the prime factorization of 24 rewritten as powers of factors? A B C D

78 Slide 78 / Which of the following is the prime factorization of 72 rewritten as powers of factors? A B C D

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81 Slide 81 / Simplify A B C D already in simplified form

82 Slide 82 / Simplify A B C D already in simplified form

83 Slide 83 / Simplify A B C D already in simplified form

84 Slide 84 / Simplify A B C D already in simplified form

85 Slide 85 / Which of the following does not have an irrational simplified form? A B C D

86 Slide 86 / 180 Simplifying Roots of Variables Divide the index into the exponent. The number of times the index goes into the exponent becomes the power on the outside of the radical and the remainder is the power of the radicand.

87 Slide 87 / 180 Simplifying Roots of Variables What about the absolute value signs? An Absolute Value sign is needed if the index is even, the starting power of the variable is even and the answer is an odd power on the outside. Examples of when absolute values are needed:

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91 Slide 91 / Simplify A B C D

92 Slide 92 / Simplify A B C D

93 Slide 93 / 180 Operations with Radicals Return to Table of Contents

94 Slide 94 / 180 Addition and Subtraction Return to Table of Contents

95 Slide 95 / 180 Operations with Radicals To add and subtract radicals they must be like terms. Radicals are like terms if they have the same radicands and the same indexes. Like Terms Unlike Terms

96 Slide 96 / Identify all of the pairs of like terms A B C D E F

97 Slide 97 / 180 To add or subtract radicals, only the coefficients of the like terms are combined.

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99 Slide 99 / Simplify A B C D Already Simplified

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101 Slide 101 / Simplify A B C D Already Simplified

102 Slide 102 / Simplify A B C D Already Simplified

103 Slide 103 / 180 Some irrational radicals will not be like terms, but can be simplified. In theses cases, simplify then check for like terms.

104 Slide 104 / Simplify A B C D Already in simplest form

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107 Slide 107 / Simplify A B C D Already in simplest form

108 Slide 108 / Which of the following expressions does not equal the other 3 expressions? A B C D

109 Slide 109 / 180 Multiplying Roots Return to Table of Contents

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111 Slide 111 / Multiply A B C D

112 Slide 112 / 180 Multiplying Square Roots After multiplying, check to see if radicand can be simplified.

113 Slide 113 / Simplify A B C D

114 Slide 114 / Simplify A B C D

115 Slide 115 / Simplify A B C D

116 Slide 116 / Simplify A B C D

117 Slide 117 / 180 Multiplying Polynomials Involving Radicals 1) Follow the rules for distribution. 2)Be sure to simplify radicals when possible and combine like terms.

118 Slide 118 / Multiply and write in simplest form: A B C D

119 Slide 119 / Multiply and write in simplest form: A B C D

120 Slide 120 / Multiply and write in simplest form: A B C D

121 Slide 121 / Multiply and write in simplest form: A B C D

122 Slide 122 / Multiply and write in simplest form: A B C D

123 Slide 123 / 180 Division: Rationalizing the Denominator Return to Table of Contents

124 Slide 124 / 180 Rationalizing the Denominator Mathematicians don't like radicals in the denominators of fractions. When there is one, the denominator is said to be irrational. The method used to rid the denominator is termed "rationalizing the denominator". Which of these has a rational denominator? Rational Denominator Irrational Denominator

125 Slide 125 / 180 If a denominator needs to be rationalized, start by finding its conjugate. A conjugate is another polynomial that when the conjugate and the denominator are multiplied, no more irrational term. The conjugate for a monomial with a square root is the same square root. Example has a conjugate of. Why? Because The conjugate of a binomial with square roots is the opposite operation between the terms. Example has a conjugate of. Why? Because

126 Slide 126 / 180 Can you find a pattern for when a binomial is multiplied by its conjugate? Example Example Example Do you see a pattern that let's us go from line 1 to line 3 directly? (term 1) 2 - (term 2) 2

127 Slide 127 / What is conjugate of? A B C D

128 Slide 128 / What is conjugate of? A B C D

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130 Slide 130 / What is conjugate of? A B C D

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132 Slide 132 / 180 The goal is to rationalize the denominator without changing the value of the fraction. To do this multiply the numerator and denominator by the same exact value. Examples:

133 Rationalize the Denominator: Slide 133 / 180 The original x in the radicand had an odd signs? power. Why no absolute value

134 Rationalize the Denominator: Slide 134 / 180

135 Slide 135 / Simplify A B C D Already simplified

136 Slide 136 / Simplify A B C D Already simplified

137 Slide 137 / Simplify A B C D Already simplified

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140 Slide 140 / 180 Rationalizing n th roots of monomials Remember that, given an n th root in the denominator, you will need to find the conjugate that makes the radicand to the n th power. Examples:

141 Slide 141 / Rationalize A B C D

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143 Slide 143 / Rationalize A B C D

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145 Slide 145 / 180 Complex Numbers Return to Table of Contents

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148 Examples Slide 148 / 180

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150 Slide 150 / Simplify A B C D

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153 Slide 153 / Simplify A B C D

154 Slide 154 / 180 Higher order i's can be simplified down to a power of 1 to 4, which can be simplified into i, -1, -i, or 1. i i 2 i 3 i 4 i 5 =i 4 i i 6 = i 4 i 2 i 7 = i 4 i 3 i 8 = i 4 i 4 i 9 = i 4 i 4 i i 10 = i 4 i 4 i 2 i 11 = i 4 i 4 i 3 i 12 = i 4 i 4 i 4 i 13 = i 4 i 4 i 4 i i 14 = i 4 i 4 i 4 i 2 i 15 = i 4 i 4 i 4 i 3 i 16 = i 4 i 4 i 4 i i raised to a power can be rewritten as a product of i 4 's and an i to the 1 st to the 4 th. Since each i 4 = 1, we need only be concerned with the non-power of 4.

155 Slide 155 / 180 To simplify an i without writing out the table say i 87, divide by 4. The number of times 4 goes in evenly gives you that many i 4 's. The remainder is the reduced power. Simplify. Example: Simplify

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157 Slide 157 / Simplify A i B -1 C -i D 1

158 Slide 158 / Simplify A i B -1 C -i D 1

159 Slide 159 / Simplify A i B -1 C -i D 1

160 Slide 160 / Simplify A i B -1 C -i D 1

161 Slide 161 / 180 Complex Numbers Recall: Operations, such as addition and division, can be done with i. Treat i like any other variable, except at the end make sure i is at most to the first power. Use the following substitutions:

162 Examples: Slide 162 / 180

163 Slide 163 / 180 Examples (in the complex form the real term comes first)

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165 Slide 165 / Simplify: A B C D

166 Slide 166 / Simplify: A B C D

167 Slide 167 / Simplify: A B C D

168 Slide 168 / Simplify: A B C D

169 Slide 169 / Simplify: A B C D

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171 Simplify Slide 171 / 180 Answers

172 Slide 172 / Simplify A B C D

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174 Slide 174 / Simplify A B C D

175 Slide 175 / 180 Dividing by i When dividing by a binomial with i, use the difference of squares to find the conjugate. Example:

176 Slide 176 / 180 Simplify: Answers

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178 Slide 178 / Simplify: A B C D

179 Slide 179 / Simplify: A B C D

180 Slide 180 / Simplify: A B C D