First of all: welcome to our community!
Second, my math may be rusty, but I think there is no reason why, for any number x, (x ^ 2) ^ -2 should give 1.
julia> x = 11//10
11//10
julia> x ^ 2
121//100
julia> (x ^ 2) ^ -2
10000//14641
julia> float((x ^ 2) ^ -2)
0.6830134553650707
I think you are thinking about multiplying the square root of x with x squared.
julia> (x^2) * (x ^ -2)
1//1