We will start off this chapter by looking at integer
exponents. In fact, we will initially
assume that the exponents are positive as well.
We will look at zero and negative exponents in a bit.
Let’s first recall the definition of exponentiation with
positive integer exponents. If a is any number and n is a positive integer then,
So, for example,
We should also use this opportunity to remind ourselves
about parenthesis and conventions that we have in regards to exponentiation and
parenthesis. This will be particularly
important when dealing with negative numbers.
Consider the following two cases.
These will have different values once we evaluate them. When performing exponentiation remember that
it is only the quantity that is immediately to the left of the exponent that
gets the power.
In the first case there is a parenthesis immediately to the
left so that means that everything in the parenthesis gets the power. So, in this case we get,
In the second case however, the 2 is immediately to the left
of the exponent and so it is only the 2 that gets the power. The minus sign will stay out in front and
will NOT get the power. In this case we
have the following,
We put in some extra parenthesis to help illustrate this
case. In general they aren’t included
and we would write instead,
The point of this discussion is to make sure that you pay
attention to parenthesis. They are
important and ignoring parenthesis or putting in a set of parenthesis where
they don’t belong can completely change the answer to a problem. Be careful.
Also, this warning about parenthesis is not just intended for
exponents. We will need to be careful
with parenthesis throughout this course.
Now, let’s take care of zero exponents and negative integer
exponents. In the case of zero exponents we have,
Notice that it is required that a not be zero. This is
important since 
is not defined. Here is a quick example of this property.
We have the following definition for negative
exponents. If a is any non-zero number and n
is a positive integer (yes, positive) then,
Can you see why we required that a not be zero? Remember that
division by zero is not defined and if we had allowed a to be zero we would have gotten division by zero. Here are a couple of quick examples for this
definition,
Here are some of the main properties of integer
exponents. Accompanying each property
will be a quick example to illustrate its use.
We will be looking at more complicated examples after the properties.
Properties
- Example :
- Example :
- Example :
- Example :
- Example :
- Example :
- Example :
- Example
:
- Example
:
- Example :
- Example :
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Notice that there are two possible forms for the third
property. Which form you use is usually
dependent upon the form you want the answer to be in.
Note as well that many of these properties were given with
only two terms/factors but they can be extended out to as many terms/factors as we need. For example, property 4 can be extended as
follows.
We only used four factors here, but hopefully you get the
point. Property 4 (and most of the other
properties) can be extended out to meet the number of factors that we have in a
given problem.
There are several common mistakes that students make with
these properties the first time they see them.
Let’s take a look at a couple of them.
Consider the following case.
In this case only the b
gets the exponent since it is immediately off to the left of the exponent and
so only this term moves to the denominator.
Do NOT carry the a down to the
denominator with the b. Contrast this with the following case.
In this case the exponent is on the set of parenthesis and
so we can just use property 7 on it and so both the a and the b move down to
the denominator. Again, note the
importance of parenthesis and how they can change an answer!
Here is another common mistake.
In this case the exponent is only on the a and so to use property 8 on this we
would have to break up the fraction as shown and then use property 8 only on
the second term. To bring the 3 up with
the a we would have needed the
following.
Once again, notice this common mistake comes down to being
careful with parenthesis. This will be a
constant refrain throughout these notes.
We must always be careful with parenthesis. Misusing them can lead to incorrect answers.
Let’s take a look at some more complicated examples now.
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Example 1 Simplify
each of the following and write the answers with only positive exponents.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
(f) [Solution]
Solution
Note that when we say “simplify” in the problem statement
we mean that we will need to use all the properties that we can to get the
answer into the required form. Also, a
“simplified” answer will have as few terms as possible and each term should
have no more than a single exponent on it.
There are many different paths that we can take to get to
the final answer for each of these. In
the end the answer will be the same regardless of the path that you used to
get the answer. All that this means
for you is that as long as you used the properties you can take the path that
you find the easiest. The path that
others find to be the easiest may not be the path that you find to be the
easiest. That is okay.
Also, we won’t put quite as much detail in using some of
these properties as we did in the examples given with each property. For instance, we won’t show the actual
multiplications anymore, we will just give the result of the multiplication.
(a)
For this one we will use property 10 first.
Don’t forget to put the exponent on the constant in this
problem. That is one of the more
common mistakes that students make with these simplification problems.
At this point we need to evaluate the first term and
eliminate the negative exponent on the second term. The evaluation of the first term isn’t too
bad and all we need to do to eliminate the negative exponent on the second
term is use the definition we gave for negative exponents.
We further simplified our answer by combining everything
up into a single fraction. This should
always be done.
The middle step in this part is usually skipped. All the definition of negative exponents
tells us to do is move the term to the denominator and drop the minus sign in
the exponent. So, from this point on,
that is what we will do without writing in the middle step.
[Return to Problems]
(b)
In this case we will first use property 10 on both terms
and then we will combine the terms using property 1. Finally, we will eliminate the negative
exponents using the definition of negative exponents.
There are a couple of things to be careful with in this
problem. First, when using the
property 10 on the first term, make sure that you square the “-10” and not
just the 10 (i.e. don’t forget the
minus sign…). Second, in the final
step, the 100 stays in the numerator since there is no negative exponent on
it. The exponent of “-11” is only on
the z and so only the z moves to the denominator.
[Return to Problems]
(c)
This one isn’t too bad.
We will use the definition of negative exponents to move all terms
with negative exponents in them to the denominator. Also, property 8 simply says that if there
is a term with a negative exponent in the denominator then we will just move
it to the numerator and drop the minus sign.
So, let’s take care of the negative exponents first.
Now simplify. We
will use property 1 to combine the m’s
in the numerator. We will use property
3 to combine the n’s and since we
are looking for positive exponents we will use the first form of this
property since that will put a positive exponent up in the numerator.
Again, the 7 will stay in the denominator since there
isn’t a negative exponent on it. It
will NOT move up to the numerator with the m. Do not get excited if
all the terms move up to the numerator or if all the terms move down to the
denominator. That will happen on
occasion.
[Return to Problems]
(d)
This example is similar to the previous one except there
is a little more going on with this one.
The first step will be to again, get rid of the negative exponents as
we did in the previous example. Any
terms in the numerator with negative exponents will get moved to the
denominator and we’ll drop the minus sign in the exponent. Likewise, any terms in the denominator with
negative exponents will move to the numerator and we’ll drop the minus sign
in the exponent. Notice this time,
unlike the previous part, there is a term with a set of parenthesis in the
denominator. Because of the
parenthesis that whole term, including the 3, will move to the numerator.
Here is the work for this part.
[Return to Problems]
(e)
There are several first steps that we can take with this
one. The first step that we’re pretty
much always going to take with these kinds of problems is to first simplify
the fraction inside the parenthesis as much as possible. After we do that we will use property 5 to
deal with the exponent that is on the parenthesis.
In this case we used the second form of property 3 to
simplify the z’s since this put a
positive exponent in the denominator.
Also note that we almost never write an exponent of “1”. When we have exponents of 1 we will drop
them.
[Return to Problems]
(f)
This one is very similar to the previous part. The main difference is negative on the
outer exponent. We will deal with that
once we’ve simplified the fraction inside the parenthesis.
Now at this point we can use property 6 to deal with the
exponent on the parenthesis. Doing
this gives us,
[Return to Problems]
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Before leaving this section we need to talk briefly about
the requirement of positive only exponents in the above set of examples. This was done only so there would be a
consistent final answer. In many cases
negative exponents are okay and in some cases they are required. In fact, if you are on a track that will take
you into calculus there are a fair number of problems in a calculus class in
which negative exponents are the preferred, if not required, form.