Copyright © 1999 by the McGraw-Hill Companies, Inc. Barnett/Ziegler/Byleen College Algebra, 6 th Edition Chapter One Basic Algebraic Operations

NNatural Numbers1, 2, 3,... ZIntegers..., –2, –1, 0, 1, 2,... QRational Numbers–4, 0, 8, –3 5, 2 3, 3.14, – __ IIrrational Numbers2,  3 7, RReal Numbers–7, 0, 3 5, –2 3, 3.14, –,  The Set of Real Numbers 1-1-1

Subsets of Real Numbers Natural numbers (N) Negative Integers Zero Integers (Z) Noninteger rational numbers Rational numbers (Q) Irrational numbers (I) Real numbers (R) N  Z  Q  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:(x +y ) + z = x + ( y + z ) Commutative:x +y =y +x Identity:0 +x =x + 0 = x Inverse:x + (– x ) = (– x ) + x =

Basic Real Number Properties Multiplication Properties Closure:xy is a unique element in R.. Associative: Commutative:xy =yx Identity:(1) x = x (1) = x Inverse: X       1 x =       1 x x = 1x  0 Combined Property Distributive: x ( y +z ) =xy +xz ( x + y ) z = +yz ( xy ) z = x ( yz )

Foil Method FOIL FirstOuterInnerLast Product  (2x – 1)(3x + 2) = 6x 2 +4x – 3x – 2 Special Products 1.(a – b)( a +b ) = a 2 – b 2 2.(a +b) 2 =a 2 + 2ab +b 2 3.(a – b) 2 =a 2 – 2ab +b

1. 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. u 3 – v 3 = (u – v )( u 2 +uv + v 2 ) Difference of Cubes 5. u 3 + v 3 = ( u + v )( u 2 –uv + v 2 ) Sum of Cubes u uv + v 2 = ( u +v) 2 Special Factoring Formulas 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.Forn a positive integer: a n =a ·a · … ·a n factors of a 2.Forn = 0, a 0 = 1a  is not defined 3.Forn a negative integer, a n = 1 a –n a  0 1.a m a n =a m + n 2. () a nm =a mn 3.(ab) m =a m b m 4.       a b m = a m b m b  0 5. a m a n =a m–n = 1 a n–m a  0 Exponent Properties Definition of a n 1-5-7

For n a natural number and b a real number, b 1/n is the principal nth root of b defined as follows: 1. If n is even and b is positive, then b 1/n represents the positive nth root of b. 2. If n is even and b is negative, then b 1/n does not represent a real number. 3. If n is odd, then b 1/n represents the real nth root of b (there is only one) /n = 0 Definition of b 1/n Rational Exponent Property For m and n natural numbers and b any real number (except b cannot be negative when n is even): 1-6-8

Rational Exponent/ Radical Conversions Properties of Radicals 1-7-9

Simplified (Radical) Form