The Zero power!!!!

Well.. this is going to be a short post...

Ummm..
what is zero raised to the power zero?
Yes, it's not defined..
Lets begin with simplest of all things..
what is the value we get when any real number (except 0 itself) is raised to the power 0?

The simple answer is.... 1!!!!

but how do we get this value?
and why can't this be true for zero raised to power zero/

We will see it in some time. first, lets clear up some basics...

Let n be a natural number....

then..

n^2 (said as n raised to the power n) = n*n
n^3 = n*n*n
.
.
.
.
n^m (where m is any other natural number) = n*n*n*n*n*n............ (m times)

all of this we all know...
When I came along these in school, there was a line written (or the teacher told us - i don't remember) that -
Anything (a real number except zero) raised to the power zero is 'taken' as 1.

I always thought that there must be something which gave us this value but no one ever explained it to me. The fact, that people did not tell me made me think. What i hate is though, it took me a hell lot of time just to think such a simple thing...

Lets see...

consider the expression ..

n^5/n^4

it can also be written as -
n^(5-4) = n^1

now i think you got my point..

n^0 = n^(m-m) = n^m/n^m = 1.....
for example -
let, n = 5  (any real number can be taken)
and m = 3 (any real number can be taken)
then,
5^0 = 5^(3-3) = 5^3/5^3 = 125/125 = 1...

Now, we can also see that why 0^0 is not defined..

put n = 0
let m be any real number..
then, we know that -
0^m = 0
now, doing as above-
0^0 = 0^(m-m) = 0^m/0^m = 0/0 
Which is not defined (anything divided by zero (even zero itself) is not defined)

So, now we have the answer to both the above questions!!!