irrational to an irrational power can be rationalIrrationalToAnIrrationalPowerCanBeRational2003-06-24 05:24:522004-01-31 05:57:31Resultirrational\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}\PMlinkescapeword{property}
Let $A=\sqrt{2}^{\sqrt{2}}$.
If $A$ is a rational number, then it has the required property.
If $A$ is an irrational number, let $B = A^{\sqrt{2}}$,
then $B=\sqrt{2}^2 = 2$ is a rational.
Hence an irrational number to an irrational power can be a rational number.
(In fact, it follows from the \PMlinkid{Gelfond-Schneider Theorem}{3952} that $A$ is transcendental, and therefore irrational.)