# 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 |
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Canonical name | IrrationalToAnIrrationalPowerCanBeRational |

Date of creation | 2013-03-22 13:42:35 |

Last modified on | 2013-03-22 13:42:35 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 10 |

Author | Koro (127) |

Entry type | Result |

Classification | msc 11A99 |

Related topic | TechniquesInMathematicalProofs |