# 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.)

Title irrational to an irrational power can be rational IrrationalToAnIrrationalPowerCanBeRational 2013-03-22 13:42:35 2013-03-22 13:42:35 Koro (127) Koro (127) 10 Koro (127) Result msc 11A99 TechniquesInMathematicalProofs