irrational to an irrational power can be rational

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 http://planetmath.org/node/3952Gelfond-Schneider Theorem that $A$ is transcendental, and therefore irrational.)