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Barnett/Ziegler/Byleen College Algebra, 6th Edition Chapter One Basic Algebraic Operations Copyright © 1999 by the McGraw-Hill Companies, Inc. The Set of Real Numbers N Natural Numbers 1, 2, 3, . . . Z Integers . . . , –2, –1, 0, 1, 2, . . . Rational Numbers __ –3 2 –4, 0, 8, 5 , 3 , 3.14, –5.2727 Q I R Irrational Numbers Real Numbers 2, –7, 0, 3 3 7 , 1.414213 . . . –2 – 5 , 3 , 3.14, 0.333 , 1-1-1 Subsets of Real Numbers Natural numbers (N) Zero Integers (Z) Rational numbers (Q) Negative Integers Noninteger rational numbers N Z Q Real numbers (R) Irrational numbers (I) R 1-1-2 Basic Real Number Properties Let R be the set of real numbers and let x, y, and z be arbitrary elements of R. Addition Properties Closure: x + y is a unique element in R. Associative: Commutative: Identity: Inverse: (x + y ) + z = x + (y + z ) x +y =y +x 0 +x = x + 0 =x x + (–x ) = (–x ) + x = 0 1-1-3-1 Basic Real Number Properties Multiplication Properties Closure: Associative: xy is a unique element . in R . ( xy ) z = x ( yz ) Commutative: Identity: xy = yx (1)x = x(1) = x Inverse: X 1 1 x = x x = 1 x 0 Combined Property Distributive: x ( y+ z ) = xy + xz ( x +y ) z = xz + yz 1-1-3-2 Foil Method F (2x – 1)(3x + 2) O I L First Outer Inner Last Product Product Product Product = – – 6x2 + 4x 3x 2 Special Products 2 2 1. (a – b)(a + b) = a – b 2. (a + b)2 = a2 + 2ab + b2 3. (a – b)2 = a2 – 2ab + b2 1-2-4 Special Factoring Formulas 1. 2 u 2 + 2 uv+ v = ( u + v)2 Perfect Square 2. u 2 – 2uv + v 2 = (u – v)2 Perfect Square 3. u 2 – v 2 = (u – v)(u + v) Difference of Squares 4. 2 u 3 – v 3 = (u – v)(u2 + uv + v ) Difference of Cubes 5. 3 2 2 u 3 + v = (u + v)(u – uv + v ) Sum of Cubes 1-3-5 The Least Common Denominator (LCD) The LCD of two or more rational expressions is found as follows: 1. Factor each denominator completely. 2. Identify each different prime factor from all the denominators. 3. Form a product using each different factor to the highest power that occurs in any one denominator. This product is the LCD. 1-4-6 Definition of an 1. For n a positive integer: Exponent Properties an = a · a · … · a n factors of a 2. For n = 0 , 2. ( a n) m = a mn a0 = 1 a 0 00 is not defined 3. For n a negative integer, 1 an = –n a 1. a m a n = a m + n a 0 3. (ab)m = a m bm am am 4. = m b b b0 am 1 5. n = a m–n = n–m a a a0 1-5-7 Definition of b1/n For n a natural number and b a real number, b1/n is the principal nth root of b defined as follows: 1. If n is even and b is positive, then b1/n represents the positive nth root of b. 2. If n is even and b is negative, then b1/n does not represent a real number. 3. If n is odd, then b1/n represents the real nth root of b (there is only one). 4. 01/n = 0 Rational Exponent Property For m and n natural numbers and b any real number (except b cannot be negative when n is even): m/n b = (bm )1/n ( b1/n ) m 1-6-8 n b , nth-Root Radical For n a natural number greater than 1 and b a real number, we define be the principal nth root of b; that is, n If n = 2, we write b in place of 2 n b to b = b1/n b. Rational Exponent/ Radical Conversions For m and n positive integers (n > 1), and b not negative when n is even, 1. 2. n n (bm)1/n = n bm bm/n = n (b1/n)m = ( b)m xn = x xy = n x n x nx 3. y =n y n y Properties of Radicals 1-7-9 Simplified (Radical) Form 1. No radicand (the expression within the radical sign) contains a factor to a power greater than or equal to the index of the radical. (For example, x5 violates this condition.) 2. No power of the radicand and the index of the radical have a common factor other than 1. (For example, 6 x4 violates this condition.) 3. No radical appears in a denominator. y (For example, violates this condition.) x 4. No fraction appears within a radical. 3 (For example, 5 violates this condition.) 1-7-10