we know that when any number is raised by zero we get 1.can you prove practically and theoritically

 

I think that an exponent of zero being equal to 1 is an empirical choice that allows a function like a^x to be continuous at x = 0.

If you are familiar with infinite power series then you know that the series for e^x yields the value 1 for x = 0; and because all the terms in x vanish at x = 0, the result is exact.


This argument is hueristic and lacks rigor but it's the best I can do in my bathrobe.

 

There are many possible proofs for this, but one that springs to mind is:

Take: x^0

And (strangely): 0 = 0 - 0

So we can say: x^0 = x^(0 - 0)

From the laws of indices: a^(m - n) = (a^m)/(a^n)

We can express: x^(0 - 0) = (x^0)/(x^0)

Therefore for any value of x, anything divided by itself is equal to 1.

QED

Dave

 

Your proof is very clear, Dave.