Property (1) says that when you multiply, you add exponents .
Property (2) says that when you raise to powers , you multiply exponenst .
It is essential to completely understand these properties since much of what we
do with more complicated
exp onents depend on these two properties. In fact, however we define other types
of exponents, we want these
two properties to be satisfied. Otherwise, all computations would have to depend
on the type of exponents
we are using.
Exponents that are not positive integers are much more difficult to
understand. For instance, nothing
about the definition for xn where n is a positive integer would tell you what
should mean. In fact, if you
were to ask your ten year old little sister what you get when you take x
times itselfzero times ,
the response would probably be “Zero”.
However, if property (1) is to be satisfied, we would have the following:
The only way this can happen is if we have the following:
Now what should be done about an exponent that is a
negative integer ? Again, the idea that an
exponent simply means to multiply something by itself a certain number of times
would not apply in this
case. What would it mean to multiply x by itself a minus five times?
We have to go back to property (1) again to see the fol lowing would have to
This leads to the following: Definition:
5. Verify the following property:
6. Show that
Now we have definitions of exponents which are integers,
whether they be positive, negative or zero.
What do we do with rational exponents?
We now rely on property (2) to see the following would have to be true:
We now recall that
re presents the number that, when multiplied by itself n times, equals x.
So we have the following:
Now we apply property (2) again to see the following:
This leads to our definition that works for all rational
or, more simply:
Great care must be taken in working with exponents and
radicals . For instance, the following statement
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