Barnett/Ziegler/Byleen
Precalculus: Functions & Graphs, 4th Edition
Appendix A
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 ,

A-1-114
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
A-1-115
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
A-1-116-1
Basic Real Number Properties
Multiplication Properties
Closure:
xy is a unique element. in R .
Associative:
( xy ) z = x ( yz )
Commutative:
xy= yx
Identity:
(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
A-1-116-2
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
1. (a – b)(a + b) = a
–
2
b
2. (a + b)2 = a2 + 2ab + b2
3. (a – b)2 = a2 – 2ab + b2
A-2-117
Special Factoring Formulas
1.
u 2 + 2 uv
+ v 2 = ( 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.
u 3 – v 3 = (u – v)(u 2 + uv + v 2)
Difference of Cubes
5.
u 3 + v 3 = (u + v)(u2 – uv + v 2)
Sum of Cubes
A-3-118
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.
A-4-119
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
am
am
4.   = m
b
b
b0
am
1
5. n = a m–n = n–m
a
a
a0
A-5-120
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
=  ( b m )1/n
( b1/ n ) m

A-6-121
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 m
1/n
m
(b
)
=
(
b)

xn = x
xy =
n
x
n x nx
3.
y =n
y
n
y
Properties of Radicals
A-7-122
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.)
A-7-123