Why does everything to the 0 power 1?

Answer 1

The multiplicative law of indices states that

xa * xb = xa+b

Now, if you put b = 0 in that equation you get

xa * x0 = xa+0

But a+0 = a so the right hand side is simply xa

Which means, the equation becomes

xa * x0 = xa

This is true for any x.

That is, multiplying any number by x0 leaves it unchanged.

By the identity property of multiplication, there is only one such number and that is 1.

So x0 must be 1.