How do you write 1 as a power in three different ways and why?

Answer 1

For the purposes of this answer we will use the convention that x^n means xⁿ

Any non-zero number to the zero power is 1

we get this from the idea that (x^n)(x^m) is x^(n+m)

if n=-m, then n+m = 0 and x^ m is 1/(x^n) and (x^n)(x^m) (x^n)/(x^n)=1 so x^0 = 1

1 raised to any positive power is 1

1 raised to any negative power is 1

We can thus represent 1 as

x^0 = 1

1^n = 1

1 ^(-n) = 1

(where n>0 Note that n need not be an integer since all roots of 1 are also 1)