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Can you evaluate the value of $2^{\sqrt{2}}$

I am not sure, if there is some trick to evaluate its value because it was asked in an interview.

But can we atleast comment if it is rational or irrational.

Obviously a rational raised to the power of irrational, can be either rational or an irrational number. So unless we do something we cannot simply comment about it nature. I am not getting any idea to proceed futher.

Bill Dubuque
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GraduateStudent
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  • Related: https://math.stackexchange.com/q/2077399/42969. – Martin R Nov 18 '19 at 12:48
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    Are you needing an approximation? You can try $2^{\sqrt 2} = 2 \times 2^{0.4} \approx 2 \times 1.3 = 2.6 $ Easiest one I can think for a quick answer in an interview. – Someone Nov 18 '19 at 12:51
  • Knowing that $\ln2\approx0.69\approx1/\sqrt2$, a very quick estimate is $e\approx2.7$. –  Nov 18 '19 at 13:00
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    It's not clear what you're asking here since it seems like you have a few questions muddled together. We can numerically find the value of $2^{\sqrt{2}}$ by many methods of approximation, as the other comments have suggested. And we can use an advanced theorem to prove that it must be transcendental and irrational. But value and irrationality have fairly little to do with each other. e.g. we know the value of $\gamma$ to billions of decimals but are no closer to finding out whether it's irrational. – Jam Nov 18 '19 at 13:05

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There isn't, no, you just need to use a calculator. It's an interesting number, but not one you can really write in another way.

By the Gelfond–Schneider theorem, it's a transcendental number.

It's also famously used in a much less advanced proof that some irrational $a,\,b$ satisfy $a^b\in\Bbb Q$. Take either $a=\sqrt{2},\,b=2\sqrt{2}$ if $2^\sqrt{2}$ is rational, or $a=2^\sqrt{2},\,b=\sqrt{2}$ if it's not.

(Actually, we usually use $b=\sqrt{2}$ in the first case or $a=\sqrt{2}^\sqrt{2}$ in the second, but it's a similar argument either way.)

J.G.
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