# Algebra Rules Review Sheet Page 7

REVIEW OF ALGEBRA
7
EXAMPLE 14
18
s18
(a)
s9
3
(b)
sx
2
y
sx
2
sy
x sy
2
s2
Notice that
sx
2
x
because
s1
indicates the positive square root.
Absolute
Value.)
(See
In general, if is a positive integer,
n
n
n
x
a
means
x
a
s
If is even, then
n
a
0
and
x
0
.
3
Thus
3
8
2
because
2
8
, but
4
8
and
6
8
are not deﬁned. The follow-
s
s
s
ing rules are valid:
n
a
a
s
n
n
n
s
ab
s
a s
b
b
n
b
s
3
3
3
3
3
EXAMPLE 15
s
x
4
s
x
3
x
s
x
3
s
x
xs
x
To rationalize a numerator or denominator that contains an expression such as
, we multiply both the numerator and the denominator by the conjugate radical
sa
sb
sa
sb
. Then we can take advantage of the formula for a difference of squares:
(
)(
) (
)
2
(
)
2
sa
sb
sa
sb
sa
sb
a
b
sx
4
2
EXAMPLE 16
Rationalize the numerator in the expression
.
x
SOLUTION
We multiply the numerator and the denominator by the conjugate radical
4
2
:
sx
4
2
4
2
4
2
x
4
4
sx
sx
sx
(
)
x
x
4
2
x
4
2
sx
sx
x
1
(
)
x
sx
4
2
sx
4
2
EXPONENTS
Let be any positive number and let be a positive integer. Then, by deﬁnition,
a
n
n
1.
a
a a
a
n factors
0
2.
a
1
1
n
3.
a
n
a
1 n
a
n
a
4.
s
m
(
)
m n
n
m
n
a
s
a
s
a
m is any integer