Download R.8 - Gordon State College

Sullivan Algebra and
Trigonometry: Section R.8
nth Roots, Rational Exponents
Objectives of this Section
• Work with nth Roots
• Simplify Radicals
• Rationalize Denominators
• Simplify Expressions with Rational Exponents
The principal nth root of a real number a,
symbolized by n a is defined as follows:
n
a  b means a  b
n
where a > 0 and b > 0 if n
is even and a, b are any
real numbers if n is odd
n
a  a,
if n is odd
n
a  a,
if n is even
n
n
Examples:
3
81  9
because 9 2  81
 27  3 because - 3  27
3
Examples:
5
12  12
5
  12    12
6
5 5
6
  5   5  5
8
z  z
5
5
6
6
8
Properties of Radicals
n
ab  a b
n
n
n
n
a
a
n
b
b
n
a   a
mn
m
a
n
mn
a
m
Simplify:
4
32 x 
5
4
16 x 2 x
 16 x
4
4
44
2x
 2 x 4 2x
Simplify:
8 x  3 x 50 x
3
 2  4  x  x  3x 25 2  x
2
= 2 x 2 x - 15 x 2 x = - 13 x 2 x
If a is a real number and n > 2 is an
integer, then
a
1
 a
n
n
provided
n
a exists.
Note that rational exponents are equivalent to
radicals. They are a different notation to
express the same concept.
Example:
8
1
3

3
82
If a is a real number and m and n
are integers containing no common
factors with n > 2, then
a
m
n
 a   a
provided
Example: 25
- 5
2
m
n
n
n
m
a exists.
1 -5
2
( )
= 25
=
(
- 5
25
)
1
1
=5 = 5 =
3125
5
- 5
Example:
 27 x 
4

1
3
3
  27 x   27 x x
4
3
  3x  x
3
 3 x  x
3
3
3
3
 3x  x
3 3
When simplifying expressions with rational
exponents, we can utilize the Laws of
Exponent.
a a a
m n
mn
a 
m n
a
mn
m
ab  a b
a
1
mn

a

if
a

0
n
nm
a
a
n
n
 a  a if b  0
 b bn
n
n n
Simplify each expression. Express the
answer so only positive exponents occur.
1
1 2
5
 3x 2 y
 3x 2  2 3 

 

2

  1 15 
3
 x y   y


1
1

4 2


2
3

3 x 


1
4 2
5
 y 
 

1
2
1

4
3 2
 3x
 4 
 5
 y 
1
4 2
5

 y 
 
1
3 2  x

1
4 2
3



y
2
5
3x
2

3
2
3y 5
2
3x 3
Write the following expression as a single
quotient in which only positive exponents
and / or radicals appear:



1
1 2
2
2
2x x  4
 x 4
2
2
x 4
x


2
4
  2 x x
1
2
2
x 4
2

1
2
 2x  x

3
4 x

2
x



2
4
  2 x x
1
2

x 4
2 x  8x  x


4 x
3
2
3
x 4
2
2
 
1
2
3
x 4
2
x  8x
3

 

x 4
2


3
2

x x 8

2
x 4
2

3
2