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,
35 = 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 =
243
Properties
Example 1 Simplify each
of the following and write the answers with only positive exponents.
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.
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.
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.
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.
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.
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.
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,
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.
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