You do an excellent job of using algebra to solve this problem. You seemed to be working towards a known answer (you gave no justification for why you made some of the steps). I suspect that this would likely have been solved by "inspired guesswork" rather than the algebra. Something like ... x^(-x^x) => (x^x^x)^-1 = 2^√2 = 2^(2^(1/2)) hints (i.e. I guess) x is a power of 2 and the ^-1 implies a negative power of 2. Clearly 2^-1 won't work so try 2^-2. It works!
