Also see https://math.andrej.com/2009/12/28/constructive-gem-irrational-to-the-power-of-irrational-that-is-rational/ https://math.andrej.com/2009/12/28/construc

Also see https://math.andrej.com/2009/12/28/constructive-gem-irrational-to-the-power-of-irrational-that-is-rational/ https://math.andrej.com/2009/12/28/constructive-gem-irration... for a similar proof using 2^(log_2 3)