Luttrell 2012
106
Name: ______________________
Date: _____
7e –Simplifying Radicals
One of the properties of e
x
ponents that was not discussed before was that of fractional e
x
ponents.
As you can tell in the e
x
ample below, fractional e
x
ponents are another way of writing radicals.
x
x
a
b
a
b
=
81
81
3
27
3
4
3
4
3
=
=
=
Before manipulating radicals,
y
ou’ll need to understand the pieces to the radical. In the
e
x
pression
x
a
b
the
b
is the
root index
. It says how many of the same number is being
multiplied together to get
x
a
. The √ is the radical sign; it implies what operation needs to be
performed. The line over the
x
a
is the
vinculum
; it is a fanc
y
name for parentheses. So
4
can
be reduced to 2 because the same two numbers that multipl
y
to get 4 is 2. Note that for square
roots, the root inde
x
is usuall
y
dropped.
144
reduces to 12. But what happens when the
number is not a perfect square? You simplif
y
the
radicand
(e
x
pression inside the radical) so it
contains no factors that are perfect squares. For n
th
roots, you want no factors that are n
th
powers.
E
x
ample A:
24
4 6
4
6
2
6
2 6
2
2
=
⋅ =
⋅
=
=
E
x
ample B:
146
2 73
146
=
⋅
=
E
x
ample C:
108
4 27
4 9 3
2 3 3
6 3
=
⋅
=
⋅ ⋅ = ⋅
=
E
x
ample D:
56
2 28
2 2
2 7
2 14
=
⋅
=
⋅ ⋅ ⋅ =
(
)
Another way to simplify a radicand is to make a factor tree and look for pairs. Better yet is to
use prime factorization.
Simplif
y
e
x
actl
y
the following:
1.
24
2.
75
3.
96
4.
102
5.
144
6.
225
7.
625
8.
525
