Well,there is a mathematical prove but I didn't see it here.
a :a =a =a
The result of dividing two numbers that are the same is 1,so a :a =1 That's why a =1 but there is a rule when a is 0. 0 and 0 where b is a negative number aren't defined and it is better to leave them instead writing a number because it can be an error.
When you divide a number by itself, each with their own exponent, you subtract the exponent (to simplify it) ; example 6^5/6^3 = 6^2 ....
OK, so if you get that, then do: 6^5/6^5 .... It equals 6^0 correct? Because you subtract the exponent 5's to get the exponent zero.
But what does dividing any number by itself give you? IT gives you 1, therefore, a^b/a^b=a^0=1
Well, I guess if that property (as I have described) had not been discovered or pieced together by someone, then we'd be looking at a^0 similar to the way that we look at a/0 ... With absolute chaos and infinite impossiblity... If you really think about it... So it has to be something simple like this to solve for anything to exponent zero .