why it is so--- a^0=1?

a^y x a^o = a^(y + o) = a^y
a^o x a^y = a^(y + o) = a^y

Thus a^o = 1

It's easy to understand if you think about it in terms of "things"

"a" is something. 0 is nothing.

So when you calculate something to the power of nothing that means you can only be left with the something. This is different than if you calculated a^1 because doing that calculation just represents "a" as itself. So if a=7 we know that there are seven numbers in "a". But when you calculate a^0 you are representing "a" against nothing, and the only thing that can be represented against nothing is something.

So given the statement a^0, "a" is the only thing of value in that statement which means there is only one thing of value. So "a" just represents itself.

"a" could actually equal any number or number of numbers but if we were to find that out we would need to look inside of "a". But given the statement a^0 we only care that "a" represented against nothing is one thing, itself. Anything by itself is equal to one.

a = 10

a^3/a^3
= 10^3/10^3
= 10^(3 - 3) (= 1000/1000)
= 10^0 (= 1)
= 1

∴ a^0 = 1

This is a guess but:

a^0 = (a^n)(a^ -n) = a^n/a^n = a/a = 1.

any thing which have zero power is equal to 1