Rational Exponents

Rational Exponents

2.   
                
             
            
               
                
             
Review of Exponential Rules

3. Simplifying Expressions with Rational Exponents
Radicals can also be expressed as a rational (or fractional) 
power of an expression.   It will sometimes be easier to 
use this new method of expressing a radical to simplify a 
radical expression.   
                    
When you see a radical 
expression, 
you can convert it to a 
fractional power.
                                       
n
bb
n
1
=

4. Example 1 Write each expression in Radical Form
6
1
x)1
6
x
3
2
)2m
3 2
m

5. Example 2Write each radical using Rational Exponents
5
b)1
5
1
b
3 75
6)2 yx 3
7
3
5
3
1
6yx

6. Example 3 Evaluate Each Expression
2
1
49)1
-
7
1

7. For any nonzero Real number a and 
any integers m and n, with n > 1
nmmn
a)a(a
n
m
==
Rational Exponents

8.                                                                  
Notice:   The index of the radical becomes the 
denominator of the rational power, and the exponent of the 
radicand (expression inside the radical) becomes the 
numerator.
(2) 
(3) (1)
Look at these examples:
root
power
xx
rootpower
=

9. Rule Example
                                                                         
              
                                 
               
                                     
Remember the Rules of Exponents?
They are still valid for rational exponents!!!

10. Example 4Evaluate 
23
)125(-
25
3
2
)125(-

11. Example 5Evaluate 
2
3
49
36
-
÷
ø
ö
ç
è
æ
216
343

12. Example 6 Simplify
( )( )
5
1
2
3
a7a3-
10
17
a21-

13.                                                                             
                                                    
Check out how these problems are done 
using the rules of exponents:
Evaluate:
Evaluate:

14. Simplify each expression completely
=

15. Expression with Rational Exponents
An expression with Rational Exponents is simplified when:
1.  It has no negative exponents
2.  It has no Fractional Exponents in the denominator
3.   It is not a Complex Fraction
4.  The index of any remaining radical is the least 
number possible.

16. Simplifying radicals is often 
easier using rational exponents.
Look at this "rational" example,
solved two ways.     ==>
 
    
                                                                          
Solved by Rationalizing the
Denominator
Solved by Using
Rational Exponents
Simplify:  
3
3
3

17. Example 7Simplify
( )
3
1
4
3
2
yx8
-
3
2
4
1
2
1
yx xy
yx
2
3
1
4
3
=

18. Example 8Simplify
7
4
n
-
n
n
7
3

19. Example 9Simplify
4
1
3
2
n
n
n
n
12
17
=
12
5
n=

20. Change to Same Base
Multiply 
Exponents 
Add Exponents
Example 8Simplify
Subtract 
Exponents
4
2
5
4
=

21. Example 9 Simplify
4a2
1a3a2
)x(
xx
-
+--
×
8a2
2a
x
x
-
-
=
= x
(a – 2) – (2a – 8)
= x
-a + 6

22. Example 10
1y
1y
2
1
2
1
-
+
Simplify
1
12
2
1
-
++
y
yy