The
n
th root of a quotient is the
quotient of the
n
th
roots.
The
n
th root of a power is the
power of the
n
th root.
When
n
is odd, the
n
th root of
a
raised to the
n
th power is
a.
When
n
is even, the
n
th root of
a
raised to the
n
th power is the
absolute value of
a.
2
4
x
4
= ƒ
x
ƒ
2
3
x
3
=
x
2
3
8
2
=
A
1
3
8
B
2
=
(2)
2
=
4
A
4
81
16
=
1
4
81
1
4
16
=
3
2
1
3
16
=
1
3
8
#
1
3
2
=
2
1
3
2
66
CHAPTER 0
Prerequisites and Review
EXAMPLE 4
Simplifying Radicals
Simplify:
a.
b.
Solution:
a.
b.
since
x
0
Ú
2
4
32
x
5
=
2
4
16
#
2
#
x
4
#
x
=
1
4
16
#
1
4
2
#
2
4
x
4
#
1
4
x
=
2
ƒ
x
ƒ
1
4
2
x
=
2
x
1
4
2
x
2
3

24
x
5
=
2
3
(

8)(3)
x
3
x
2
=
1
3

8
#
1
3
3
#
2
3
x
3
#
2
3
x
2
=

2
x
2
3
3
x
2
2
4
32
x
5
2
3

24
x
5

2
s
x
s
2
s
ƒ
x
ƒ
s
Combining Like Radicals
We have already discussed properties for multiplying and dividing radicals. Now we focus
on combining (adding or subtracting) radicals. Radicals with the same index and radicand
are called
like radicals
. Only like radicals can be added or subtracted.
EXAMPLE 5
Combining Like Radicals
Combine the radicals if possible.
a.
b.
c.
d.
Solution (a):
Use the distributive property.
Eliminate the parentheses.
Solution (b):
None of these radicals are alike.
The expression is in simplified form.
Solution (c):
Write the radicands as
products with a factor of 5.
The square root of a product is the
product of square roots.
Simplify the square roots of perfect squares.
All three radicals are now like radicals.
Use the distributive property.
Simplify.
Solution (d):
None of these radicals are alike because
they have different indices. The expression
is in simplified form.
■
YOUR TURN
Combine the radicals.
a.
b.
5
1
24

2
1
54
4
1
3
7

6
1
3
7
+
9
1
3
7
1
4
10

2
1
3
10
+
3
1
10

1
5
(3
+
2

6)
1
5
3
1
5
+
2
1
5

6
1
5
3
1
5
+
2
1
5

2(3)
1
5
3
1
5
+
1
4
#
1
5

2
1
9
#
1
5
3
1
5
+
1
20

2
1
45
=
3
1
5
+
1
4
#
5

2
1
9
#
5
2
1
5

3
1
7
+
6
1
3
=
5
1
3
4
1
3

6
1
3
+
7
1
3
=
(4

6
+
7)
1
3
1
4
10

2
1
3
10
+
3
1
10
3
1
5
+
1
20

2
1
45
2
1
5

3
1
7
+
6
1
3
4
1
3

6
1
3
+
7
1
3
6
s
■
Answer:
a.
b.
4
1
6
7
1
3
7
0.6
Rational Exponents and Radicals
67
s
1
Rationalizing Denominators
When radicals appear in a quotient, it is customary to write the quotient with no radicals
in the denominator. This process is called
rationalizing the denominator
and involves
multiplying by an expression that will eliminate the radical in the denominator.
For example, the expression
contains a single radical in the denominator. In a
case like this, multiply the numerator and denominator by an appropriate radical expression,
so that the resulting denominator will be radical free:
If the denominator contains a sum of the form
, multiply both the numerator and
the denominator by the
conjugate
of the denominator,
, which uses the difference
of two squares to eliminate the radical term. Similarly, if the denominator contains a
difference of the form
, multiply both the numerator and the denominator by the
conjugate of the denominator,
. For example, to rationalize
, take the
conjugate of the denominator, which is
:
In general we apply the difference of two squares:
Notice that the product does not contain a radical. Therefore, to simplify the expression
multiply the numerator and denominator by
:
The denominator now contains no radicals:
A
1
a

1
b
B
(
a

b
)
1
A
1
a
+
1
b
B
#
A
1
a

1
b
B
A
1
a

1
b
B
A
1
a

1
b
B
1
A
1
a
+
1
b
B
A
1
a
+
1
b
B A
1
a

1
b
B
=
A
1
a
B
2

A
1
b
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 Summer '17
 juan alberto
 Square Roots, Nth root