Negative Rational Exponent in Radical Expressions

5.7 Rational Exponents
Fraction Exponents

Radical expression and Exponents
By definition of Radical Expression.
The index of the Radical is 3.
5
125
125
5 3
3

 so

How would we simplify this
expression?
What does the fraction exponent do to the
number?
The number can be written as a Radical
expression, with an index of the
denominator.
2
1
9

2
9

The Rule for Rational Exponents
4
64
64 3
3
1
1


 n
n
b
b

Write in Radical form


2
1
6
1
m
a

Write in Radical form
m
m
a
a


2
1
6
6
1

Write each Radical using Rational
Exponents


w
b
5

Write each Radical using Rational
Exponents
2
1
5
1
5
w
w
b
b



What about Negative exponents
Negative exponents make inverses.
7
1
49
1
49
2
1
2
1




What if the numerator is not 1
Evaluate
5 2
5
2
32
32 

Evaluate
  5 10
5 2
5
5 2
5
2
2
2
32
32



Evaluate
 
4
2
2
2
32
32
2
5 10
5 2
5
5 2
5
2




For any nonzero real number b,
and integer m and n
Make sure the Radical express is real, no
b<0 when n is even.
 m
n
n m
n
m
b
or
b
b 

Simplify each expression
No fraction in the denominators.
3
2
7
4
7
1


x
y
y

Simplify
3
1
6
1
3
6
2
16
2
16


Simplify
3
1
6
4
3
1
6
1
4
3
1
6
1
3
6
2
2
2
)
2
(
2
16
2
16




Simplify
3
3
1
3
1
3
2
3
1
3
2
3
1
6
4
3
1
6
1
4
3
1
6
1
3
6
2
2
2
2
2
2
2
2
2
)
2
(
2
16
2
16









Simplify
6
4
6
1
6 4
4
4 x
x 

Simplify
  6
4
6
1
2
6
4
6
1
6 4
2
4
4
x
x
x



Simplify
 
6
4
6
2
6
4
6
1
2
6
4
6
1
6 4
2
2
4
4
x
x
x
x




Simplify
 
3
2
3
1
6
4
6
2
6
4
6
1
2
6
4
6
1
6 4
2
2
2
4
4
x
x
x
x
x





Simplify
 
  3 2
3
1
2
3
2
3
1
6
4
6
2
6
4
6
1
2
6
4
6
1
6 4
2
2
2
2
2
4
4
x
x
x
x
x
x
x







Negative Rational Exponent in Radical Expressions

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