Is $\sqrt{3}^\sqrt{5}$ rational or irrational?

One way is to let $x$=$\sqrt{3}^\sqrt{5}$ and then calculate $antilog \ (log (\sqrt 3) \times \sqrt(5))$ which gives irrational number.

But is there a way to check it without calculators..

  • 3
    $\begingroup$ It's not even algebraic. $\endgroup$ – Asaf Karagila Dec 26 '14 at 16:40
  • $\begingroup$ I'm not sure why the form $10^{\frac{\log(3) \sqrt{5}}{2}}$ is any more likely to be irrational than what you've written. A computer can't tell you a number is irrational by direct computation (though a sophisticated program could work out the proof of some theorem.) $\endgroup$ – Ian Dec 26 '14 at 16:53

This is a direct application of the Gelfond–Schneider theorem. Interestingly, I don't think it's easier to prove that this number is irrational than to prove it is transcendental!


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