You already know of one
relationship between exponents and radicals: the appropriate radical will
"undo" an exponent, and the right power will "undo"
a root. For example:

But there is another relationship
(which, by the way, can make computations like those above much simpler):
For the square (or "second") root, we can write it as the one-half
power, like this:

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...or:

The cube (or "third")
root is the one-third power:

The fourth root is the
one-fourth power:

The fifth root is the one-fifth
power; and so on.

Looking at the first examples,
we can re-write them like this:

You can enter fractional
exponents on your calculator for evaluation, but you must remember to
use parentheses. If you are trying to evaluate, say, 15^{(4/5)},
you must put parentheses around the "4/5",
because otherwise your calculator will think you mean "(15
^{4}) ÷ 5".

Fractional exponents allow
greater flexibility (you'll see this a lot in calculus), are often easier
to write than the equivalent radical format, and permit you to do calculations
that you couldn't before. For instance:

Whenever you see a fractional
exponent, remember that the top number is the power, and the lower number
is the root (if you're converting back to the radical format). For instance:

By the way, some decimal
powers can be written as fractional exponents, too. If you are given something
like "3^{5.5}",
recall that 5.5
= ^{11}/_{2},
so:

3^{5.5}
= 3^{11/2}

Generally, though, when
you get a decimal power (something other than a fraction or a whole number),
you should just leave it as it is, or, if necessary, evaluate it in your
calculator. For instance, 3
^{pi}
, where pi
is the number approximately equal to 3.14159,
cannot be simplified or rearranged as a radical.

A technical point:
When you are dealing with these exponents with variables, you might have
to take account of the fact that you are sometimes taking even roots.
Think about it: Suppose you start with the number –2.
Then:

In other words, you put
in a negative number, and got out a positive number! This is the official
definition of absolute
value:

(Yeah, I know: they never
told you this, but they expect you to know somehow, so I'm telling you
now.) So if they give you, say, x^{3/6},
then x
had better not be negative, because x^{3}
would still be negative, and you would be trying to take the sixth root
of a negative number. If they give you x^{4/6},
then a negative x becomes positive (because of the fourth power) and is
then sixth-rooted, so it becomes |
x |^{2/3}
(by reducing the fractional power). On the other hand, if they give you
something like x^{4/5},
then you don't have to care whether x
is positive or negative, because a fifth root doesn't have any problem
with negatives. (By the way, these considerations are irrelevant if your
book specifies that you should "assume all variables are non-negative".)

A technology
point: Calculators
and other software do not compute things the way people do; they
use pre-programmed algorithms. Sometimes the particular method the
calculator uses can create difficulties in the context of fractional
exponents.

For instance, you
know that the cube root of –8
is –2,
and the square of –2
is 4, so (–8)^{(2/3)}
= 4. But some
calculators return a complex value or an error message, as is the
case with one of my graphing calculators:

If you enter "=(–8)^(2/3)"
into a cell, the Microsoft "Excel" spreadsheet returns
the error "#NUM!".

Some calculators
and programs will do the computations as expected, as displayed
at right from my other graphing calculator:

The difference
has to do with the pre-programmed calculating algorithms. These
algorithms generally try to do the computations in ways which require
the fewest "operations", in order to process what you've
entered as quickly as possible. But sometimes the fastest method
isn't always the most useful, and your calculator will "choke".

Fortunately,
you can get around the problem: by splitting the numerator and denominator
of the fractional power, your calculator should arrive at the correct
value:

As you
can see above, it didn't matter if I first took the cube root of
negative eight and then squared, or if I first squared and then
cube-rooted; either way, the calculator returned the proper value
of "4".