Doesn't the first argument have a logic error, in that x^(1-1) = x^1x^(-1) has a caveat for x not equal to 0, so it shouldn't prove anything for x = 0. In other

Doesn't the first argument have a logic error, in that x^(1-1) = x^1x^(-1) has a caveat for x not equal to 0, so it shouldn't prove anything for x = 0. In other words, it just says x^0 = 1 for any x =/= 0. Using Abstract Algebra, I think 0^0 = 1 is completely accurate. The power function (y^x) could be defined to be the amount you times (x times) you apply the operation between the y on the identity element. In our usual numbers that looks like y(y(y...(y1)...)). When x is negative y becomes the multiplicative inverse of y and everything else remains the same.