Fractional Exponents
If n is a positive integer , then
 is the
 root of a .
If a is positive, it is the positive number b such that
If a is negative, then:
1. If n is odd, is the
negative number b such that
2. If n is even, is undefined.
is also written
 |
Example.
 since
 is the same as
 Note that
is not “±3”.
 since 
 since
 is undefined.
Note, however, that
Exponentiation takes precedence over negation. |
If m and n are positive integers ,
 means
This makes sense, since
and this should equal
if the rule for multiplying
exponents is to hold in this case.
Equivalently ,
In other words, you can do the root and the power in either order .
 involves an nth root, so it may be
positive, negative, or undefined. |
Example.
 |
Example. You can use a calculator to
compute roots and powers. How you do this depends on what kind of
calculator you have . For example,
Roots of negative numbers can present a problem ; some calculators will
return a complex number , or
give an error message. You can fix things by figuring out the sign of
the result beforehand. Then make the
base positive for your calculator and fix the sign at the end.
For example,  is an odd root of a
negative number, so it’s negative. Knowing this, I use the
calculator to compute 
Therefore,
( |
Example. There is some ambiguity here. For
example, 
 since
But  should be undefined, since
I’m taking an even root of a negative number. I will avoid this
problem by always expressing fractional exponents in lowest terms .
|
The rules I gave earlier for working with integer
exponents work with fractional exponents — with
certain exceptions for even roots.
 |
Example.
 |
Example.
 |
However,
 is not the same as x.
Why?  and
 is always nonnegative. But x could
be negative: For example,
 but x = −3.
In this case, 
In fact, if n is an even integer,
(Of course, I can drop the absolute values if I know x is nonnegative.) |
| Example.
 |
| Example. If x and y are nonnegative,
 |
| Example.
 |
Example. |
| Example. Assuming that x and y are
nonnegative,
 |
Example. |
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