Why do any number raised to 0th power is always 1?

Answer 1

Look at this sequence:

103 = 1000

102 = 100

101 = 10

100 = ???

What would you expect the result to be? On the left, the exponent is reduced by one each time, on the right, the number is reduced by a factor of 10, so the logical continuation is 1. Replace the base (10) with any other number (except zero), and you get the same result.

Defining x0 as 1 makes several algebraic laws consistent, for example, xa + xb = xa+b. If x0 were anything else than 1, it would have to be treated as a special case in this law - or the law would only make sense for positive exponents.

Using a very similar reasoning, 10-1 = 1/10, 10-2 = 1/100, etc.