Algebra II Radical Equations

Transcription

1 1

2 Algebra II Radical Equations

3 Table of Contents: Graphing Square Root Functions Working with Square Roots Irrational Roots Adding and Subtracting Radicals Multiplying Radicals Rationalizing the Denominator Cube Roots nth Roots Rational Exponents Solving Radical Equations Complex Numbers click on the topic to go to that section If review is needed before or during this unit click on the link below. Fundamental Skills of Algebra (Supplemental Review) Click for Link 3

4 Graphing Square Root Functions Return to Table of Contents 4

5 Inverse of Squares Recall the Inverse of Squares... The inverse of is...but the result is not a function. 5

6 Inverse of Squares The domain of y = x 2 is restricted to x 0, so that the inverse will be a function. Domain: [0, ) Range: [0, ) Domain: [0, ) Range: [0, ) 6

7 Steps of Graphing a Square Root Function 1. Find the domain of the square root function by setting the expression under the square root (the radicand) greater than or equal to zero and solve for x. 2. Choose 3 values for x in the domain that will give you perfect squares under the root. Plug into the original function to find y. 3. Plot your points and graph. x y Teacher Notes 7

8 Parent Functions The function is one of the parent functions. Use this fact to help you anticipate the graph or find the function from the graph. Remember transformations... Parent Domain: [0, ) Range: [0, ) Domain: [-3, ) Range: [0, ) Domain: [3, ) Range: [0, ) 8

9 Parent Functions And... Parent Domain: [0, ) Range: [0, ) Domain: [0, ) Range: [3, ) Domain: [0, ) Range: [-3, ) 9

10 1 Which is the graph of the function? A B C D 10

11 2 Which is the graph of the function? A B C D 11

12 3 Which is the equation of the graph? A B C D 12

13 4 What is the domain of: A B C D 13

14 5 Find the range of: A B C D 14

15 6 What is the domain of: A B C D 15

16 7 Find the range of: A B C D 16

17 Parent Functions Remember what happens when you have af(x) or f(bx)... Parent Domain: [0, ) Range: [0, ) Domain: [0, ) Range: (-, 0] Domain: [0, ) Range: [0, ) 17

18 Parent Functions And... Parent Domain: [0, ) Range: [0, ) Domain: (-, 0] Range: [0, ) Domain: (-, 0] Range: (-, 0] 18

19 8 Which is the graph of the function? A B C D 19

20 9 Which is the graph of the function? A B C D 20

21 10 Which is the equation of the graph? A B C D 21

22 11 Which is the equation of the graph? A B C D 22

23 Square Root Functions What happens when you combine the transformations? Why is this one only moved to the right 2? Teacher Notes 23

24 Square Root Functions In order to see how much the graph moves horizontally, any coefficient must be factored out. Rewrite the following by factoring out the coefficient: Teacher Notes 24

25 Square Root Functions In order to see how much the graph moves horizontally, any coefficient must be factored out. Rewrite the following by factoring out the coefficient: Teacher Notes 25

26 12 How far does the graph move horizontally? Use a negative (-) to indicate a move left. 26

27 13 How far does the graph move horizontally? Use a negative (-) to indicate a move left. 27

28 14 How far does the graph move horizontally? Use a negative (-) to indicate a move left. 28

29 15 How far does the graph move horizontally? Use a negative (-) to indicate a move left. 29

30 16 How far does the graph move horizontally? Use a negative (-) to indicate a move left. 30

31 Graphing Functions To graph the function: 1. Anticipate the graph using transformations. 2. Set the radical expression 0 and solve to find the domain. 3. Choose 3 values in the domain that will give you perfect squares under the radical. Use a t-table to find y's. 4. Plot the points and graph. Teacher Notes 31

32 Graph the function: Graphing Functions 1. Anticipate the graph using transformations. 2. Set the radical expression 0 and solve to find the domain. 3. Choose 3 values in the domain that will give you perfect squares under the radical. Use a t-table to find y's. 4. Plot the points and graph. Teacher Notes 32

33 Graphing Functions Graph the function: 1. Anticipate the graph using transformations. 2. Set the radical expression 0 and solve to find the domain. 3. Choose 3 values in the domain that will give you perfect squares under the radical. Use a t-table to find y's. 4. Plot the points and graph. Teacher Notes 33

34 Graphing Functions Graph the function: 1. Anticipate the graph using transformations. 2. Set the radical expression 0 and solve to find the domain. 3. Choose 3 values in the domain that will give you perfect squares under the radical. Use a t-table to find y's. 4. Plot the points and graph. Teacher Notes 34

35 Graphing Functions Graph the function: 1. Anticipate the graph using transformations. 2. Set the radical expression 0 and solve to find the domain. 3. Choose 3 values in the domain that will give you perfect squares under the radical. Use a t-table to find y's. 4. Plot the points and graph. Teacher Notes 35

36 Working with Square Roots Return to Table of Contents 36

37 Square Roots Recall... * Teacher Notes 37

38 Square Roots All of these numbers can be written with a square. Since the square is the inverse of the square root, they "undo" each other. 38

39 17 What is? 39

40 18 Find: 40

41 19 What is? 41

42 20 What is? 42

43 21 Find: 43

44 Variables What happens when you have variables in the radicand? To take the square root of a variable rewrite its exponent as the square of a power. 44

45 Variables IMPORTANT: When taking the square root of variables, remember that answers must be positive. Even powered answers, like the last page, will be positive even if the variables are negative. The same cannot be said if the answer has an odd power. When you take a square root and the answer has an odd power, put the result inside an absolute value symbol. 45

46 22 Simplify: A B C D 46

47 23 Simplify: A B C D 47

48 24 Simplify: A B C D 48

49 25 Simplify: A B C D 49

50 Square Roots of Fractions For square roots of fractions, take the square root the numerator (top) and denominator (bottom) separately. Teacher Notes 50

51 26 A B C D no real solution 51

52 27 A B C D no real solution 52

53 28 A B C D no real solution 53

54 29 A B C D no real solution 54

55 30 A B C D no real solution 55

56 Irrational Roots Return to Table of Contents 56

57 Simplifying Radicals is said to be a rational number because there is a perfect square that equals the radicand. If a radicand cannot be made into a perfect square, the root is said to be irrational, like. 57

58 Simplifying Radicals The commonly accepted form of a radical is called simplest radical form. To simplify numbers that are not perfect squares, start by breaking the radicand into factors and then breaking the factors into factors and so on until only prime numbers are left. This is called prime factorization. 58

59 Prime Factorization Examples of Prime Factorization: Note: There is maybe more than one way to break a part a number, but the prime factorization will always be the same. 59

60 31 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) 60

61 32 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) 61

62 33 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) 62

63 34 Which of the following is the prime factorization of 24 rewritten as powers of factors? A B C D 63

64 35 Which of the following is the prime factorization of 72 rewritten as powers of factors? A B C D 64

65 Simplifying Non-Perfect Square Radicands Find the prime factorization of the radicand, group prime factors to make perfect squares. Simplify. This is simplest radical form. Teacher Notes 65

66 36 Simplify: A B C D already in simplified form 66

67 37 Put in simplest radical form: A B C D already in simplified form 67

68 38 Put in simplest radical form: A B C D already in simplified form 68

69 39 Simplify: A B C D already in simplified form 69

70 40 Which of the following is not an irrational number? A B C D 70

71 Simplifying Radicals If there is a number, or expression, on the outside of the root remember that it is held together by multiplication. To simplify, put the root in simplest radical form and multiply. 45 Teacher Notes 71

72 41 Put in simplest radical form: A B C D 72

73 42 Simplify: A B C D 73

74 43 Put in simplest radical form: A B C D 74

75 44 Put in simplest radical form: A B C D 75

76 45 Put in simplest radical form: A B C D 76

77 Simplifying Radicals with Absolute Values The same process goes for variables, but absolute value signs need to be included where appropriate. Absolute value symbols are required when the initial exponent is even and the exponent after taking the root is odd. If the initial exponent is odd, you will not need absolute values. 77

78 Simplifying Radicals with Absolute Values Examples: Teacher Notes 78

79 46 Simplify: A B C D 79

80 47 Put in simplest radical form: A B C D 80

81 48 Simplify: A B C D 81

82 49 Put in simplest radical form: A B C D 82

83 50 Put in simplest radical form: A B C D 83

84 Adding and Subtracting Radicals Return to Table of Contents 84

85 Adding and Subtracting Radicals *Note: When adding or subtracting radicals, you do not add or subtract the radicands (the inside). Consider: 85

86 Adding and Subtracting 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 86

87 Adding and Subtracting Radicals An index indicates what root you are taking. Just like square roots undo squares, cube roots undo cubes, fourth roots undo powers of four, fifth roots undo powers of 5, etc... This concept will be studied more in depth later in the unit. 87

88 51 Identify all of the pairs of like terms: A B C D E F 88

89 Adding and Subtracting Radicals To add or subtract radicals, only the coefficients of the like terms are combined - just like 3x + 4x = 7x. Teacher Notes 89

90 Adding and Subtracting Radicals Try... Teacher Notes 90

91 Adding and Subtracting Radicals It is the same for expressions containing variables. Simplify: Teacher Notes 91

92 52 Simplify: A B C D Already Simplified 92

93 53 Simplify: A B C D Already Simplified 93

94 54 Simplify: A B C D Already Simplified 94

95 55 Simplify: A B C D Already Simplified 95

96 56 Simplify: A B C D Already Simplified 96

97 Adding and Subtracting Radicals Some irrational radicals will not be like terms, but could be put in simplest radical form. In theses cases, simplify, then collect any like terms. Teacher Notes 97

98 Adding and Subtracting Radicals The same goes for expressions containing variables. Try: Teacher Notes 98

99 57 Simplify: A B C D Already in simplest form 99

100 58 Simplify: A B C D Already in simplest form 100

101 59 Simplify: A B C D Already in simplest form 101

102 60 Simplify: A B C D Already in simplest form 102

103 61 Simplify: A B C D Already simplified 103

104 62 Simplify: A B C D Already simplified 104

105 Multiplying Radicals Return to Table of Contents 105

106 Multiplying Radicals When multiplying radicals, you may multiply radicands. Consider

107 Multiplying Radicals Whole number times whole number and radical times radical. Never multiply a whole number and radical! Leave all answers in simplest radical form. Teacher Notes 107

108 Multiplying Radicals Examples: Teacher Notes 108

109 Examples: Multiplying Radicals Teacher Notes 109

110 63 Multiply: A B C D 110

111 64 Simplify: A B C D 111

112 65 Simplify: A B C D 112

113 66 Simplify: A B C D 113

114 67 Simplify: A B C D 114

115 Multiplying Polynomials with Radicals Leave all answers in simplest radical form Teacher Notes 115

116 68 Multiply and write in simplest form: A B C D 116

117 69 Multiply and write in simplest form: A B C D 117

118 70 Multiply and write in simplest form: A B C D 118

119 71 Multiply and write in simplest form: A B C D 119

120 72 Multiply and write in simplest form: A B C D 120

121 Rationalizing the Denominator Return to Table of Contents 121

122 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? Teacher Notes Rational Denominator Irrational Denominator 122

123 If the denominator is a monomial, to rationalize, just multiply top and bottom of the fraction by the root part of the denominator. Examples: Rationalizing the Denominator Teacher Notes 123

124 Rationalizing the Denominator If a denominator is a binomial with a root, rationalize the denominator by multiplying top and bottom by finding its conjugate. The conjugate of a binomial is found by negating the second term of a binomial. Binomial: Conjugate: 124

125 Rationalizing the Denominator Multiplying by the conjugate turns an irrational number into a rational number. Check out what happens... Teacher Notes 125

126 Rationalizing the Denominator Do you see a pattern that let's us go from line 1 to line 3 directly? Example Example Example Teacher Notes 126

127 Rationalizing the Denominator Use conjugates to rationalize the denominators: Teacher Notes 127

128 Rationalizing the Denominator Use conjugates to rationalize the denominators: Teacher Notes 128

129 73 What is conjugate of? A B C D 129

130 74 What is conjugate of? A B C D 130

131 75 Simplify: A B C D Already simplified 131

132 76 Simplify: A B C D Already simplified 132

133 77 Simplify: A B C D Already simplified 133

134 78 Simplify: A B C D Already simplified 134

135 79 Simplify: A B C D Already simplified 135

136 80 Simplify: A B C D Already simplified 136

137 81 Simplify: A B C D Already simplified 137

138 Cube Roots Return to Table of Contents 138

139 Cube Roots If a square root cancels a square, what cancels a cube? Teacher Notes 139

140 Cube Roots is read "the cube root of 64" A cube root looks like a square root. The difference is that little 3. The 3 (or other root) is called the index of a radical. Square roots have an index of two, but it is not usually written. An index indicates how many of each number or variable would allow taking a perfect root. A cube root is looking for groups of three, just as a square root is looking for groups of two. 140

141 Cube Roots The cube root and the cube cancel each other out. Examples: Teacher Notes 141

142 Try... Cube Roots Teacher Notes 142

143 Cube Roots Notice Where as is not real. Roots of negative numbers can only be taken if the index is odd. 143

144 82 Select all of the possible radicals that have real answers. A B C D E F G H I J 144

145 83 Evaluate the radical: A 3 B 4 C 6 D 8 145

146 84 Evaluate the radical: A 3 B 4 C 6 D 8 146

147 85 Evaluate the radical: A 2 B -2 C -4 D No real answer 147

148 86 Evaluate the radical: A -1 B - 1 / 3 C 1 /3 D 1 148

149 Cube Roots Just like square roots, cube roots can also be put in simplest radical form. Instead of looking for groups of 2, just look for groups of 3! Teacher Notes 149

150 Cube Roots The techniques and methods for solving square roots of variables, fractions, and decimals also work with cube roots. No absolute value signs are needed when using an odd index. Therefore, the answers for cube roots will not require absolute values. Teacher Notes 150

151 Cube Roots Put in simplest radical form: Teacher Notes 151

152 87 Simplify: A B C D not possible 152

153 88 Simplify: A B C D not possible 153

154 89 Simplify: A B C D not possible 154

155 90 Simplify: A B C D not possible 155

156 91 Simplify: A B C D not possible 156

157 92 Put in simplest radical form: A B C D not possible 157

158 n th Roots Return to Table of Contents 158

159 Absolute Values In general,. Absolute value signs are necessary if n is even, the initial exponent is even and the variable has an odd powered exponent after taking the root. Teacher Notes 159

160 n th Roots Try... Teacher Notes 160

161 93 Simplify: A B C D 161

162 94 Simplify: A B C D 162

163 95 Simplify: A B C D 163

164 96 Simplify: A B C D 164

165 97 Simplify: A B C D 165

166 98 Simplify: A B C D 166

167 99 Simplify: A B C D 167

168 100 Simplify: A B C D 168

169 Simplest Radical Form 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. 169

170 Absolute Value Signs As always, what about 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 outside the radical is an odd power. Examples of when absolute values are needed: 170

171 n th Roots Try... Teacher Notes 171

172 101 Simplify: A B C D 172

173 102 Simplify: A B C D 173

174 103 Simplify: A B C D 174

175 104 Simplify: A B C D 175

176 105 Simplify: A B C D 176

177 Rationalizing n th roots of Monomials Remember that, given an n th root in the denominator, it will need to be rationalized. To rationalize, find the complement if the n th root that will create a perfect root in the denominator. Multiply top and bottom by the complement. Simplify. Examples: Teacher Notes 177

178 Try: Rationalizing n th roots of Monomials Teacher Notes 178

179 106 Rationalize: A B C D 179

180 107 Rationalize: A B C D 180

181 108 Rationalize: A B C D 181

182 109 Rationalize: A B C D 182

183 110 Simplify: A B C D 183

184 111 Rationalize: A B C D 184

185 112 Simplify: A C B D 185

186 Rational Exponents Return to Table of Contents 186

187 Rational Exponents Rational exponents, or exponents that are fractions, are another way to write and work with radicals. Power Root 187

188 Rational Exponents Simplify: Teacher Notes 188

189 Rational Exponents Simplify: Teacher Notes 189

190 113 Simplify: 190

191 114 Simplify: 191

192 115 Simplify: 192

193 116 Simplify: 193

194 117 Simplify: 194

195 Rational Exponents Rewrite each radical as a rational exponent in the lowest terms. Teacher Notes 195

196 Rational Exponents Rewrite each radical as a rational exponent in the lowest terms. Teacher Notes 196

197 Combining Radicals Rewrite each expression as a single radical. To combine more than one number or variable, the roots must be the same. Teacher Notes 197

198 Combining Radicals When the roots (denominators) are different, they must be made into a common number in order to create a single root. Teacher Notes 198

199 118 Find the expression that is equivalent to: A B C D 199

200 119 Find the simplified expression that is equivalent to: A B C D 200

201 120 Find the simplified expression that is equivalent to: A B C D 201

202 121 Find the simplified expression that is equivalent to: A B C D 202

203 122 Find the simplified expression that is equivalent to: A B C D 203

204 123 Simplify: A B C D 204

205 124 Write with rational exponents: A B C D 205

206 125 Find the simplified expression that is equivalent to: A B C D 206

207 126 Write the following with exponents: A B C D 207

208 Rational Exponents When working with rational exponents, follow exactly the same rules as when working with other exponents. Teacher Notes 208

209 Rational Exponents Just like other problems where you must rationalize denominators, mathematicians like to have a an integer power in the denominators. Therefore, if there is a fractional exponent in the denominator after simplifying, rationalize the denominator. Teacher Notes 209

210 127 Simplify: A B C D 210

211 128 Simplify: A B C D 211

212 129 Simplify: A B C D 212

213 130 Simplify: A B C D 213

214 131 Simplify and write as a radical: A B C D 214

215 132 Simplify. Make sure your denominator is rational. A C B D 215

216 Solving Radical Equations Return to Table of Contents 216

217 Solving Radical Equations To solve a radical equation: 1. Isolate the radical on one side of the equation. 2. Use the index to determine the power to use to eliminate the radical. 3. Raise both sides of the equation to that power. 3. Solve the resulting equation. 4. Check to see if solution is extraneous. Teacher Notes 217

218 Solving Radical Equations Example: Teacher Notes 218

219 Solving Radical Equations Example: Teacher Notes 219

220 Example: Solving Radical Equations Teacher Notes 220

221 133 Find the solution to: 221

222 134 Find the solution to: 222

223 135 Find the solution to: 223

224 136 Find the solution to: 224

225 137 Find the solution to: 225

226 Solving Radical Equations If an equation has multiple roots, move them to opposite sides of the equal sign and then solve. Teacher Notes 226

227 138 Solve the following: 227

228 139 Solve the following: 228

229 140 Solve: 229

230 141 Solve: 230

231 Complex Numbers Return to Table of Contents 231

232 Complex Numbers The square root of a negative number has no real solution, but it does have an imaginary one: An expression is complex (also called imaginary) if it has an i in it. 232

233 Complex Numbers Complex Numbers: All numbers are technically considered complex numbers. Real Numbers can be written as a + 0i - no imaginary component. Imaginary Numbers Real Numbers Rational Numbers Integers Whole Numbers Natural Numbers Irrational Numbers 233

234 Complex Numbers Why does this work? Teacher Notes 234

235 Complex Numbers Higher order i's can be simplified into i, -1, -i, or 1. If the power of i is even:...and the exponent is a multiple of 4, then it simplifies to 1....and the exponent is a multiple of 2,but not 4, then it simplifies to -1. If the power of i is odd:...factor out one i to create an even exponent. Use the rules for even exponents and leave the factored i. 235

236 Simplify the following: Complex Numbers Remember Teacher Notes 236

237 Complex Numbers Remember More examples: Teacher Notes 237

238 142 Simplify: A i B -1 C -i D 1 238

239 143 Simplify: A i B -1 C -i D 1 239

240 144 Simplify: A i B -1 C -i D 1 240

241 145 Simplify: A i B -1 C -i D 1 241

242 Complex Numbers Simplify radical expressions that have a negative by taking out i first. Then, perform the indicated operation(s). Simplify any expression that has a power of i greater than one. Teacher Notes 242

243 Examples: Complex Numbers Teacher Notes 243

244 146 Simplify: A B C D 244

245 147 Simplify: A B C D 245

246 148 Simplify: A B C D 246

247 149 Simplify: A B C D 247

248 150 Simplify: A B C D 248

249 Working with Complex Numbers Operations, such as addition, subtraction, multiplication 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. 249

250 Working with Complex Numbers s of complex numbers are left in standard form. The standard form of a complex number is a + bi. Examples of standard form of a complex number: 3-2i 0 + 3i 8 + 0i 250

251 Adding or Subtracting Complex Numbers When adding or subtracting complex numbers, collect like terms. Leave answers in standard form. Teacher Notes 251

252 Multiplying Complex Numbers When multiplying, multiply numbers, multiply i's and simplify any i with a power greater than one. Teacher Notes 252

253 Multiplying Complex Numbers When multiplying, multiply numbers, multiply i's and simplify any i with a power greater than one. Teacher Notes 253

254 Multiplying Complex Numbers Multiply and leave answers in standard form. Teacher Notes 254

255 Multiplying Complex Numbers Multiply and leave answers in standard form. Teacher Notes 255

256 151 Simplify: A B C D 256

257 152 Simplify: A B C D 257

258 153 Simplify: A B C D 258

259 154 Simplify: A B C D 259

260 155 Simplify: A B C D 260

261 Dividing with i Since i represents a square root, a fraction is not in simplified form if there is an i in the denominator. And, similar to roots, if the denominator is a monomial just multiply top and bottom of the fraction by i to rationalize. Teacher Notes 261

262 Dividing with i Simplify: Teacher Notes 262

263 Dividing with i Simplify: Teacher Notes 263

264 156 Simplify: A B C D 264

265 157 Simplify: A B C D 265

266 158 Simplify: A B C D 266

267 Rationalizing Complex Numbers If the denominator is a binomial including i, rationalize it by multiplying top and bottom by its conjugate. Remember using conjugates earlier in this unit: the conjugate of 4-3i is 4 + 3i. Example: Teacher Notes 267

268 Rationalizing Complex Numbers Simplify: Teacher Notes 268

269 159 Simplify: A B C D 269

270 160 Simplify: A C B D 270

271 161 Simplify: A C B D 271

272 162 Simplify: A C B D 272