Fractional exponents are related to roots or radicals.

If n is a positive integer, then

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.* Compute the exact values of:

(a) .

(b) .

(c) .

(d) .

(e) .

is the same as . Note that is *not*
" ".

In the last example, 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 root, so it may be positive, negative, or undefined.

* Example.* Compute the exact values of:

(a) .

(b) .

(c) .

(d) .

(e) .

* Example.* Use a calculator to approximate and .

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.* Is " " undefined, since -1 is negative and
the root is an even root?

Before considering a fractional exponent, the fraction should be reduced to lowest terms: .

The rules I gave earlier for working with integer exponents work with
fractional exponents --- with certain exceptions for *even
roots*.

* Example.* Simplify .

* Example.* Simplify .

However,

Why? , and is always nonnegative. But x could be negative: For example,

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.* Simplify .

Note that since I didn't assume x was nonnegative, the answer is not "x".

* Example.* Assuming that x and y are
nonnegative, simplify .

* Example.* Simplify .

* Example.* Assuming that x and y are
nonnegative, simplify . Write your answer using positive powers.

* Example.* Assuming that x and y are
nonnegative, simplify . Write your answer using positive powers.

* Example.* Assuming that x and y are
nonnegative, simplify . Write your answer using positive
powers.

* Example.* Assuming that x and y are
nonnegative, simplify . Write your answer using positive powers.

* Example.* Assuming that x, y, and z are
nonnegative, simplify . Write your answer using positive powers.

Copyright 2016 by Bruce Ikenaga