Fundamentals of AlgebraChu v. NguyenIntegral Exponents

Fundamentals of Algebra
Chu v. Nguyen
Integral Exponents
Exponents
If n is a positive integer (a whole number, i.e., a number
without decimal part) and x is a number, then
The number x is called the base and n is called the exponent.
The most common ways of referring to are “ x to the nth
power,”
“ x to the nth,” or “the nth power of x.”
Integral Exponents (cont.)
For any non-zero number x and a positive integer n
and
Note: is not defined
and

Rules Concerning Integral Exponents
Following are five rules in which m and n are positive integers:
Rule 1: ; for example,
Rule 2: ; for example
or

Rules Concerning Integral Exponents (Cont.)
or
or
or

Basic Rules for Operating with Fractions
Since dividing by zero is not defined, we assume that the
denominator
is not zero.
Following are the eight basic rules for operating with fractions.
Basic Rules for Operating with Fractions (cont.)

Basic Rules for Operating with Fractions (cont.)
Notes: a*b +a*x may be expressed as a(b + x)
a*b + 1 may be written as a(b + ), and
m*x – y may be expressed as m(x - )

Square Root
Generally, for a>0 , there is exactly one positive number x such
that
, we say that x is the root of a, written as
for
When n = 2, we say that x is the square root of “a” and is
denoted by
or or
For example:
or

Practices
Carrying out the following operations:
24 ; 2-2 ; 2322, ; 252-5 ; and (2x3)5
; ; ; and
3.
4.
5.
n
m
n
m
x
x
x
+
=
n

x
1
0
)
2
(
2
2
2
=
=
=
-
+
-
x
x
x
x
1
0
=
x
0
0
n
n
n
x
x
x
x
=
=
+
0

0
n
n
x
x
1
=
-
3
2
5
5
2
0
2
2
2
2
3
2
5
2
5
1
1
1
z
z
z
z
n
m
y
y
y
y

n
m
x
x
x
x
n
m
=
=
<
=
=
=
=
=
=
>
-
-
-
n
m
n
m
x
x
x
+
=
9
5
4
5
4
3

3
3
3
=
=
+
ç
ç
ç
ç
ç
è
æ
<
=
>
=
-
-
n
m
if
b
n
m
if
n
m
if
b
b
b
m
n
n
m

n
m
1
1
3
2
5
5
2
0
2
2
2
2
3
2
5
2
5
2
1
2
1
2
2
1
5
5
5
5
2
2
2
2
=
=

<
=
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=
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-
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-
n
m
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m
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2
.
2
2
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3
2
2
3
3
2
)
3
2
(
=
2
.
2

2
.
3
2
2
3
)
(
y
x
y
x
=
n
n
n
c
b
c
b
=
÷
ø
ö
ç
è
æ
5
5
5
2
3
2
3
=
÷

ø
ö
ç
è
æ
3
3
3
y
x
y
x
=
÷
÷
ø
ö
ç
ç
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æ
n
m
n
m
x
x
.
)
(
=
6
2
.
3
2

3
2
2
)
2
(
=
=
6
2
.
3
2
3
)
(
x
x
x
=
=
n
n
n
y
x
xy
=
)
(
7
2
5
*
7
5

*
2
=
7
*
5
4
*
3
7
4
*
5
3
=
b
a
b
a
b
a
-
=
-
=
-
b
a
by
ay
=
bd
ac
d
c
b

a
=
*
9
7
9
7
9
7
-
=
-
=
-
2
5
10
5
7
3
5
7
5
3
=
=
+
=
+
7
1
7
3
4
7
3

7
4
=
-
=
-
bc
ad
c
d
b
a
d
c
b
a
=
=
¸
*
d
c
a
d
c
d
a
+
=
+
d
c
a
d
c
d

a
-
=
-
5
3
10
6
4
*
5
6
*
2
4
6
*
5
2
6
4
5
2
=
=
=
=
¸
a
1
m
y
bd
bc
ad
d

c
b
a
+
=
+
12
1
24
2
24
20
18
6
*
4
5
*
4
6
*
3
6
5
4
3
-
=
-
=
-
=
-
=
-
bd

bc
ad
d
c
b
a
-
=
-
12
19
24
20
18
6
*
4
)
5
*
4
(
)
6
*
3
(
6
5
4
3
=
+
=
+
=

+
a
x
=
5
.
0
a
x
=
2
1
a
x
=
00
.
5
25
25
25
5
.
0
2
1
=
=
=
=
x
4772
.
5
30
30

30
5
.
0
2
1
=
=
=
=
z
a
x
n
=
th
n
n
a
x
=
2
³
n
3
5
4
÷
ø
ö
ç
è
æ
81
=
z

75
=
x
7
4
3
2
+
5
3
3
4
-
5
3
7
9
¸
9
7
*
5
3
Review of Fundamental Mathematics
Chu V. Nguyen
Algebraic Numbers
Each algebraic number (hence forth, referred to as a number)
consists of two components. A POSITIVE numeric value which
can be expressed as:
2, 5, 8; or in notation such as a, b, x, y, z; or

and an algebraic sign in front:
+ (plus) or – (minus)
Usually, the + (plus) sign is omitted; for examples:
+2 is often expressed as 2, or +z is written as z; and as
Basic Properties of Algebraic Numbers
Closure Properties:
The sum or product of two algebraic numbers is an algebraic
number; thus,
+a-x or a-x or, -z-a is an algebraic number.
and
(+a)*(+b), or a*b, (a)(-x), or (-z)*(-a) is also an algebraic
number.
Note: a.b may be expressed in a number of other ways, such as
(a)(b), (a)(b), a(b), (a)b or just ab.
“*” indicates “multiplication.”
Basic Properties of Algebraic Numbers (cont.)
Commutative Properties:
The sum or product of two numbers is not affected by the order
in which they are combined. That is, for numbers a and z
a + z = z + a, or a +(-z) = -z + a
a*b = b*a , or (a)(-z) = (-z)(a).
Examples:
3 + 5 = 5 + 3 = 8 or 6 + (-4) = 6 – 4 = 2.

3*4 = 4*3 =12 or 5(-4) = (-4)*5 = -20
Associative Properties:
The sum or product of three numbers is the same when the third
is combined with the first two or when the first is combined
with the last two . That is, for numbers a, y and z
(a + y) + z = a + ( y + z), or (a + y) - z = a + (y -z)
(a*y)z = a( yz), or (ay)(- z) = a[y(-z)]
Distributive Properties:
The product of a number times the sum or the difference of two
others is the same as the sum or the difference of the products
of the first number times each of the others. That is, for
numbers a, y and z
a(y+ z) = ay + az, or a(y – z) = ay + a(-z) = ay –
az.
This may also be written in the form:
(y + z)a = ya + za, or (y – z)a = y + (-z)a = ya –za.
Identity Properties:
The sum of any number x and a zero is the given number, x.
0 + x = x + 0 = x
We call zero the additive identity.
The product of any number x and 1 is the given number, x.
1*x = x*1 = x

We define 1 as the multiplicative identity.
Inverse Properties:
For each number x, there exists another number, -x, such that
the sum of x and –x is zero.
x + (-x) = 0
We define –x or (-x) as the additive inverse or opposite of x.
For each number x, different from zero, there exists another
number ,
such that the product of x and is 1.
x* = *x = 1
We call the multiplicative inverse or reciprocal of x.
Other Inverse Properties:
The opposite of the opposite of a number z is the number z
-(-z) = z
The opposite of the sum is the sum of the opposites. That is ,

for numbers q and z
-(q + z) = (-q) + (-z)
The opposite of a product of two numbers is the product of one
number times the opposite of the other. That is for two numbers
a and b
-(ab) = (-a)b = a(-b)
Properties of zero:
The product of a number p and zero is zero.
p*0 = 0*p = 0.
It is also important to note that for any two numbers x and y
If x*y = 0, then either x = 0, or y = 0 or both x = 0 and y =
0.
Subtraction and Addition
The difference 0r sum of two numbers x and z is defined as
x – z = x + (-z).
For examples,
5 - 8 = 5 + (-8) = -3.
or 5 – (-8) = 5 + 8 = 13
Alternatively, we can say that for three numbers a, b, and c
a – b = c if and only if a - c = b
Thus 5-2 = 3 because 5 - 3 = 2
Note: “if and only if” is sometime written as “iff.”
Multiplication and Division
The product 0r the quotient of two numbers a and z is defined
as

(a)(– z) or a(-z).
; z must be non-zero, since we cannot divide
by zero
Signs:
The product and the quotient of two numbers, a and z will have
a positive ( + or plus) sign if a and z have the same sign.
The product and the quotient of two numbers a and z will have
a negative ( - or minus) sign if a and z have opposite signs.
Practices
Evaluate the following expressions:
(i) 2(-3)
(ii) -5(x)
(iii) (5*6)(7)
2. Find the additive inverse for 4.
3. Find the multiplicative inverse for 2
4. Find the values for the following expressions:
(i) 8 – (-4)
(ii) 2 + (-5)
x
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K
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p
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x
1
x
1
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