Discussion
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.)
Rule 3: ; for example
or
Rule 4: ; for example
or
Rule 5: ; for example
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.
Rule 1: ; for example
Rule 2: ; for example
Rule 3: ; for example
Basic Rules for Operating with Fractions (cont.)
Rule 4: ; for example
Rule 5: ; for example
Rule 6: ; for example
Basic Rules for Operating with Fractions (cont.)
Rule 7: ; for example
Rule 8: ; for example
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
n
m
n
m
2
.
2
2
.
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
è
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