Copyright © 1999 by the McGraw-Hill Companies, Inc.

Barnett/Ziegler/ByleenCollege Algebra, 6th Edition

Chapter One

Basic Algebraic Operations

N Natural Numbers 1, 2, 3, . . .

Z Integers . . . , –2, –1, 0, 1, 2, . . .

Q Rational Numbers –4, 0, 8, –35 ,

23 , 3.14, –5.2727

__

I Irrational Numbers 2 , 37 , 1.414213 . . .

R Real Numbers –7, 0, 3

5 , –23 , 3.14, 0.333

– ,

The Set of Real Numbers

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Subsets of Real Numbers

Natural numbers (N)

Negative Integers

ZeroIntegers (Z)

Noninteger rational numbers

Rational numbers (Q)

Irrational numbers (I)

Real numbers (R)

N Z Q R

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Basic Real Number Properties

Let R be the set of real numbers and let x, y, and z be arbitraryelements 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 = 0

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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 = 1 x 0

Combined Property

Distributive: x (y + z ) = xy + xz(x + y )z = xz + yz

( xy ) z = x ( yz )

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Foil Method

F O I LFirst Outer Inner LastProduct Product Product Product

(2x

– 1)(3x + 2)

=6x2 + 4x – 3x – 2

Special Products

1. (a – b)(a + b) = a2 – b2

2. (a + b)2 = a2 + 2ab + b2

3. (a – b)2 = a2 – 2ab + b2

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1. Perfect Square

2. u2 – 2uv + v 2 = (u – v)2 Perfect Square

3. u2 – v 2 = (u – v)(u + v) Difference of Squares

4. u3 – v

3 = (u – v)(u2 + uv + v

2 ) Difference of Cubes

5. u3 + v

3 = (u + v)(u2 – uv + v

2 ) Sum of Cubes

u 2 + 2uv + v 2 = ( u + v)2

Special Factoring Formulas

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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.

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1. For n a positive integer:

an = a · a · … · an factors of a

2. For n = 0 ,

a0 = 1 a 000 is not defined

3. For n a negative integer,

an = 1

a–n a 0

1. am an = am+ n

2. ( )an m = amn

3. (ab)m = am bm

4.

a

bm

= am

bm b 0

5.am

an = am–n = 1

an–m a 0

Exponent Properties

Definition of an

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For n a natural number and b a real number,

b1/n is the principal nth root of bdefined 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

Definition of b1/n

Rational Exponent Property

For m and n natural numbers and b any real number (except b cannot be negative when n is even): bm/n =

()

()

/

/

b

b

nm

mn

1

1

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For n a natural number greater than 1 and b a real number, we define n

b tobe the principal nth root of b; that is,

nb = b1/n

If n = 2, we write b in place of 2

b .

nb , nth-Root Radical

For m and n positive integers (n > 1), and b not negative when n is even,

bm/n =

(bm)1/n = n

bm

(b1/n)m = (n

b)m

Rational Exponent/Radical Conversions

1. n

xn = x

2. n

xy = n

x n

y

3. n x

y = n

x

ny

Properties of Radicals

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1. No radicand (the expression within the radical sign) contains afactor 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 acommon factor other than 1.

(For example, 6

x4 violates this condition.)

3. No radical appears in a denominator.

(For example, yx violates this condition.)

4. No fraction appears within a radical.

(For example, 35 violates this condition.)

Simplified (Radical) Form

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