Example 1 Evaluate each of the following.
Solution
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Solution
When doing these evaluations we will do actually not 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.
So,
here is what we are asking in this problem.
Using the equivalence from the definition we can
rewrite this as,
?2 = 25
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,
So what we are asking here is what number did we raise
to the 5th power to get 32?
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?.
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,
Example 2 Evaluate each of the following.
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.
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.
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.
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.
Example 3 Simplify each of the following and write the answers with only
positive exponents.
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.
(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.
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