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Chipola College MGF 1107 5.4 The Irrational Numbers and the Real Number System_________________________ Irrational number is a real number whose decimal representation is a nonterminating, nonrepeating decimal number. Examples/ 6.01020304….. , .353553555…. , Determine whether the number is rational or irrational. 1. 4.323223222…. 2. MEMORIZE Perfect “Squares” 12 = 1 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36 72 = 49 82 = 64 92 = 81 102 = 100 112 = 121 122 = 144 132 = 169 142 = 196 152 = 225 162 = 256 172 = 289 182 = 324 192 = 361 202 = 400 3. 4.16739 16 4. 24 A. Definitions An expression of the form radicand and 3 5 a i i r is a radical expression where r is the is the index. Name the index and the radicand in the following: ( x y) abc 7 5x 2 Radical expressions can be simplified using the properties: n an a Examples: 25 = 196 (___) 2 = _____ (5) 2 = _____ 144 = _______ 49 = _________ - 36 =___ 64 = _________ Review examples - Simplify the following. 1. 4 2. 4. 81 3. - 225 361 5. 289 B. To simplify radical expressions involving factors, use the property of multiplication with ab a b radicals: Examples: 48 16 3 4 3 28 4 7 2 7 Use this property to simplify the following expressions. 1. 20 2. 60 75 4. 90 5. c. 450 d. 2232 e. 2250 3. 300 You Try: a. 98 b. 72 C. Adding and Subtracting Radical Expressions. If the terms have the same radicand just combine their coefficients. 1. 3 5 7 5 2. 2 11 7 11 3. 6 3 6 9 6 4. 3 3 3 8 3 If the terms do NOT have the same radicand, simplify the radicals and then combine those that have the same radicand. 1. 5 3 12 2. 2 5 3 20 3. 2 7 5 28 4. 13 2 2 18 5 32 You Try: a. 6 5 2 5 b. 3 98 7 2 18 c. 5 27 4 48 a b ab D. Multiplying Radical expressions. Multiply the radicands together and then simplify. 1. 3 27 You Try: a. 3 8 2. 3 7 b. 3. 6 10 5 15 4. c. 11 33 3 6 E. To simplify radical expressions involving quotients, use the property of division with radicals: a a b b (Note: This property works for any index) Use this property to simplify the following expressions. 1. 49 16 2. 8 2 3. 136 8 4. You Try: a. 8 4 b. 125 5 c. 96 2 75 3 F. If a radical remains in the denominator we “agree” to rationalize the denominator as follows: Multiply the numerator and denominator by the radical in the denominator. Rationalize the denominator. 1. 5 2 2. 5 12 3. 5 10 4. 3 3 5. 8 8 6. 3 10 b) 7 7 c) You try: a) 2 10 25 81