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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


  is the same as   Note that is not “±3”.



is undefined.

Note, however, that

Exponentiation takes precedence over negation.
If m and n are positive integers ,


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. 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


Example. There is some ambiguity here. For example,


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.


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. If x and y are nonnegative,


Example. Assuming that x and y are nonnegative,

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