# Algebra Rules Review Sheet Page 8

8
REVIEW OF ALGEBRA
11
Laws of Exponents
Let and be positive numbers and let and be any
a
b
r
s
rational numbers (that is, ratios of integers). Then
r
a
r
s
r s
r s
r s
rs
a
a
a
a
a
a
1.
2.
3.
s
a
r
r
a
a
r
r
r
4.
ab
a
b
5.
b
0
r
b
b
In words, these ﬁve laws can be stated as follows:
1.
To multiply two powers of the same number, we add the exponents.
2.
To divide two powers of the same number, we subtract the exponents.
3.
To raise a power to a new power, we multiply the exponents.
To raise a product to a power, we raise each factor to the power.
4.
To raise a quotient to a power, we raise both numerator and denominator to
5.
the power.
EXAMPLE 17
8
2
8
3 2
8
6
14
(a)
2
8
2
2
2
2
2
2
2
1
1
y
x
2
2
2
2
2
2
2
2
x
y
x
y
x
y
y
x
xy
(b)
1
1
2
2
x
y
1
1
y
x
x
y
y
x
x
y
xy
y
x y
x
y
x
xy y
x
xy
3
(
)
3 2
3 2
3
(c)
4
3
8
4
2
8
s4
s64
Alternative solution:
s4
1
1
4 3
(d)
x
4 3
3
4
s
x
x
3
4
2
3
8
4
x
y
x
x
y
x
7
5
4
z
(e)
x
y
3
4
y
z
y
z
INEQUALITIES
When working with inequalities, note the following rules.
Rules for Inequalities
1.
If
a
b
, then
a
c
b
c
.
2.
If
a
b
and
c
d
, then
a
c
b
d
.
3.
If
a
b
and
c
0
, then
ac
bc
.
4.
If
a
b
and
c
0
, then
ac
bc
.
5.
If
0
a
b
, then
1 a
1 b
.
Rule 1 says that we can add any number to both sides of an inequality, and Rule 2 says
that two inequalities can be added. However, we have to be careful with multiplication.
Rule 3 says that we can multiply both sides of an inequality by a positive
|
number, but Rule 4 says that
if we multiply both sides of an inequality by a negative num-
ber, then we reverse the direction of the inequality.
For example, if we take the inequality