If you want to stay out of trouble, then the last way you wrote it may be the best. This is because the function 
for real 
and 
is generally defined as 
, and 
and 
are generally defined by power series, which require 
where 
is a whole number. This is easy to define as "repeated multiplication", as long as you don't let 
. Then, the power series definition will imply that 
for all nonzero real 
. However, it gives no value for when 
. By limits, it's clear to see that there is no value for 
which makes 
continuous at 
, so there's no "natural" choice in that sense. In this case, the power series for 
does not imply any value for 
.
You can, of course, deal with 
at the beginning, by letting 
for all real 
(including 
). This basically chooses 
to be 
as a convention, and one upshot of this is that it simplifies the power series definition of exponential functions. But in this case, the choice of 
implies the power series for 
is 
, not the other way around.
So, no matter how you slice it, the power series for 
does not imply that 
, but rather we can choose to let 
to imply the simplified power series for 
.
If you want to stay out of trouble, then the last way you wrote it may be the best. This is because the function for real and is generally defined as , and and are generally defined by power series, which require where is a whole number. This is easy to define as "repeated multiplication", as long as you don't let . Then, the power series definition will imply that for all nonzero real . However, it gives no value for when . By limits, it's clear to see that there is no value for which makes continuous at , so there's no "natural" choice in that sense. In this case, the power series for does not imply any value for .
You can, of course, deal with at the beginning, by letting for all real (including ). This basically chooses to be as a convention, and one upshot of this is that it simplifies the power series definition of exponential functions. But in this case, the choice of implies the power series for is , not the other way around.
So, no matter how you slice it, the power series for does not imply that , but rather we can choose to let to imply the simplified power series for .
is equal to 
because multiplying by something 
times amounts to not multiplying by anything, which is the same as multiplying by 
However, at the same time, 
is an indeterminate form because if 
and 
as 
then 
can approach 
or 
or 
or any other member of 
depending on what functions 
and 
are. However, in a sense, in most cases the limit of 
will be 
One case is when 
within some "sector", i.e. it stays between two positively sloped lines in the 
-plane.
is not undefined; it is 
.
However, 
and 
does not imply tat 
, but that is a totally different story - namely that 
is called an indeterminate form.
The first term of the development is formally
which is indubitably defined to be 
for all 
. Then it is a natural convention (possibly left implicit) that this term is always 
, making the function continuous while allowing a notational convenience.
Unless he stated this convention, the author would have been more careful to write
as @mathisfun said.
Note that this doesn't tell us anything about the value of 
, if it has one.
First I want to say I’m doing algebra 1 this year and this may be covered later in math, but ever since I learned anything the the power of 0 equals 1 I’ve been confused by that. I’ve tried searching it online but I can’t find anything that explains it, everything I can find doesn’t answer the question.
From my understanding how exponents work is the exponent is how many of that number you have then you multiply them. For ex. 324 is 32 * 32 * 32 * 32. So shouldn’t for ex. 920 = 0 since that would mean you don’t have any 92s.
Edit: thanks to everyone who actually answered in stead of saying that it’s not true with 00 or just not answering.