Answer:
Typically, at this point beginning algebra texts note that all variables are assumed to be positive. If this is the case, then x in the previous example is positive and the absolute value operator is not needed. The example can be simplified as follows:
√
9
x
2
=
√
3
2
x
2
=
√
3
2
⋅
√
x
2
=
3
x
In this section, we will assume that all variables are positive. This allows us to focus on calculating nth roots without the technicalities associated with the principal nth root problem. For this reason, we will use the following property for the rest of the section:
n
√
a
n
=
a
,
if
a
≥
0
n
th root
When simplifying radical expressions, look for factors with powers that match the index.
Example 3: Simplify:
√
18
x
3
y
4
.
Solution: Begin by determining the square factors of 18,
x
3
, and
y
4
.
Make these substitutions and then apply the product rule for radicals and simplify.
Answer:
3
x
y
2
√
2
x
Example 4: Simplify:
√
4
a
5
b
6
.
Solution: Begin by determining the square factors of 4,
a
5
, and
b
6
.
