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After a small search that I did I was unable to spot any answers here.What I am trying is to prove why the $2^{\sqrt2}$ is transcendental number. I know that this probably is a closed problem and probably many people have proved it already,but I want to reach the answer by myself after doing a research on this problem and going through several books and sources and test my knowledge and my power.So what I basically need is to give me guideline to solve the problem.If my question is too general,before you close it,please give me some information to rephrase,so it won't be so general.

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2 Answers 2

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By the Gelfond–Schneider theorem, $2^{\sqrt2}$ is transcendental.

$2^{\sqrt 2}$ is called the Gelfond–Schneider constant.

See also https://math.stackexchange.com/questions/173804/deciding-whether-2-sqrt2-is-irrational-transcendental

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  • $\begingroup$ Thank you for your valuable help.Now at least I know where I can find the answer and what to look for to gain more knowledge. $\endgroup$
    – el10780
    Commented Sep 24, 2012 at 19:39
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Don't worry, your question is well-stated. Some of these simply stated questions about transcendental numbers are still unknown! I don't know about this one. But here is another: Is $\pi^\pi$ a rational number? This one was unknown 20 years ago, and I expect it is still unknown.

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  • $\begingroup$ Oh thank God.I am not experienced like you guys here.Thanks for your reply and thanks for giving me another problem to think about.(Although I do not think I will something like that.0 :) One think I can't understand though.Why they closed this question.The users who closed it claimed that it is off-topic.Off-topic compared to what?Did I misused the tagging? $\endgroup$
    – el10780
    Commented Sep 24, 2012 at 19:38
  • $\begingroup$ @el10780, this site is for research-level questions. For other questions, see math.stackexchange.com . $\endgroup$
    – lhf
    Commented Sep 25, 2012 at 2:25
  • $\begingroup$ Ok.Thank you for your respond and I apologize for my ignorance. $\endgroup$
    – el10780
    Commented Sep 25, 2012 at 5:39

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