Now that we have looked at integer exponents we need to
start looking at more complicated exponents.
In this section we are going to be looking at rational exponents. That is exponents in the form
where both m and n are integers.
We will start simple by looking at the following special
case,
where n is an
integer. Once we have this figured out
the more general case given above will actually be pretty easy to deal with.
Let’s first define just what we mean by exponents of this
form.
In other words, when evaluating we are really asking what number (in this case
a) did we raise to the n to get b. Often is called the nth root of b.
Let’s do a couple of evaluations.
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Example 1 Evaluate
each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
(f) [Solution]
Solution
When doing these evaluations, we will not actually do them directly. When first confronted
with these kinds of evaluations doing them directly is often very
difficult. In order to evaluate these
we will remember the equivalence given in the definition and use that
instead.
We will work the first one in detail and then not put as
much detail into the rest of the problems.
(a)
So, here is what we are asking in this problem.
Using the equivalence from the definition we can rewrite
this as,
So, all that we are really asking here is what number did
we square to get 25. In this case that
is (hopefully) easy to get. We square
5 to get 25. Therefore,
[Return to Problems]
(b)
So what we are asking here is what number did we raise to
the 5th power to get 32?
[Return to Problems]
(c)
What number did we raise to the 4th power to
get 81?
[Return to Problems]
(d)
We need to be a little careful with minus signs here, but
other than that it works the same way as the previous parts. What number did we raise to the 3rd
power (i.e. cube) to get -8?
[Return to Problems]
(e)
This part does not have an answer. It is here to make a point. In this case we are asking what number do
we raise to the 4th power to get -16. However, we also know that raising any
number (positive or negative) to an even power will be positive. In other words, there is no real number
that we can raise to the 4th power to get -16.
Note that this is different from the previous part. If we raise a negative number to an odd
power we will get a negative number so we could do the evaluation in the
previous part.
As this part has shown, we can’t always do these
evaluations.
[Return to Problems]
(f)
Again, this part is here to make a point more than
anything. Unlike the previous part
this one has an answer. Recall from
the previous section that if there aren’t any parentheses then only the part
immediately to the left of the exponent gets the exponent. So, this part is really asking us to
evaluate the following term.
So, we need to determine what number raised to the 4th
power will give us 16. This is 2 and
so in this case the answer is,
[Return to Problems]
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As the last two parts of the previous example has once again
shown, we really need to be careful with parenthesis. In this case parenthesis makes the difference
between being able to get an answer or not.
Also, don’t be worried if you didn’t know some of these
powers off the top of your head. They
are usually fairly simple to determine if you don’t know them right away. For instance in the part b we needed to
determine what number raised to the 5 will give 32. If you can’t see the power right off the top
of your head simply start taking powers until you find the correct one. In other words compute ,
,
until you reach the correct value. Of course in this case we wouldn’t need to go
past the first computation.
The next thing that we should acknowledge is that all of the
properties for exponents that we
gave in the previous section are still valid for all rational exponents. This includes the more general rational
exponent that we haven’t looked at yet.
Now that we know that the properties are still valid we can
see how to deal with the more general rational exponent. There are in fact two different ways of
dealing with them as we’ll see. Both
methods involve using property 2 from the previous section. For reference purposes this property is,
So, let’s see how to deal with a general rational
exponent. We will first rewrite the
exponent as follows.
In other words we can think of the exponent as a product of
two numbers. Now we will use the
exponent property shown above. However,
we will be using it in the opposite direction than what we did in the previous
section. Also, there are two ways to do
it. Here they are,
Using either of these forms we can now evaluate some more
complicated expressions
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Example 2 Evaluate
each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
We can use either form to do the evaluations. However, it is usually more convenient to
use the first form as we will see.
(a)
Let’s use both forms here since neither one is too bad in
this case. Let’s take a look at the
first form.
Now, let’s take a look at the second form.
So, we get the same answer regardless of the form. Notice however that when we used the second
form we ended up taking the 3rd root of a much larger number which
can cause problems on occasion.
[Return to Problems]
(b)
Again, let’s use both forms to compute this one.
As this part has shown the second form can be quite
difficult to use in computations. The
root in this case was not an obvious root and not particularly easy to get if
you didn’t know it right off the top of your head.
[Return to Problems]
(c)
In this case we’ll only use the first form. However, before doing that we’ll need to
first use property 5 of our
exponent properties to get the exponent onto the numerator and denominator.
[Return to Problems]
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We can also do some of the simplification type problems with
rational exponents that we saw in the previous section.
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Example 3 Simplify
each of the following and write the answers with only positive exponents.
(a) [Solution]
(b) [Solution]
Solution
(a) For this problem we will first move the exponent into the
parenthesis then we will eliminate the negative exponent as we did in the
previous section. We will then move
the term to the denominator and drop the minus sign.
[Return to Problems]
(b) In this case we will first simplify the expression inside the
parenthesis.
Don’t worry if, after simplification, we don’t have a
fraction anymore. That will happen on
occasion. Now we will eliminate the
negative in the exponent using property
7 and then we’ll use property 4
to finish the problem up.
[Return to Problems]
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We will leave this section with a warning about a common
mistake that students make in regards to negative exponents and rational
exponents. Be careful not to confuse the
two as they are totally separate topics.
In other words,
and NOT
This is a very common mistake when students first learn exponent
rules.