For example $\sqrt 7$ is irrational but $\sqrt 7$ raised to power $2$ is rational. Similarly, is it possible that $\pi$ raised to some power (say $n$) could be rational ?
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4No, pi is transcendental. – Joca Ramiro Oct 09 '19 at 12:38
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5$\pi^0 = 1$ is rational XD – glowstonetrees Oct 09 '19 at 12:40
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2so you mean integer prower right? otherwise the answer would be an immediate yes. – Seyhmus Güngören Oct 09 '19 at 12:40
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2@SeyhmusGüngören is asking for the right clarification. Otherwise,$\pi^{\ln2 /\ln \pi} =2$. – Randall Oct 09 '19 at 12:44
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@SeyhmusGüngören The transcendentality argument also applies to positive rational powers. – Arnaud D. Oct 09 '19 at 12:44
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@ArnaudD. yes, I mean just a bit more information would be good. The OP gave $2$ as an example so my most likely guess would be that he meant first of all integers. – Seyhmus Güngören Oct 09 '19 at 12:47
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For any positive rational value $q$, we have
$$\pi^{\log_{\pi}(q)} = q$$
so certainly yes, $\pi$ to some power can be rational.
If, however, you only allow integer powers, then the answer is no. In fact, even if you would allow rational nonzero powers of $\pi$, the answer is no. We know this because we know that $\pi$ is a transcendental number*, which means it is not the root of any polynomial with rational coefficients. If there would exist some rational number $r=\frac{a}{b}$ (where $a,b\in\mathbb N$) such that $\pi^r = q\in\mathbb Q$, then we would have
$$0=\left(\pi^r\right)^b-(q^a)^b = \pi^a - q^{ab}$$
which means that $\pi$ would be the solution to the equation $x^a-q^{ab}=0$, which we know is impossible.
* Proving $\pi$ is transcendental was quite a feat back when it was first done and was by no means an easy task!
5xum
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@YvesDaoust Interesting question! Don't know the answer from the top of my head though... – 5xum Oct 09 '19 at 12:59
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@5xum I'd venture to say no. But, algebraic numbers can be pretty weird. – Rushabh Mehta Oct 09 '19 at 13:22
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Here I meant that, is there any power (any complex number) upto which π is raised to get rational number? – MR. HACKER BUDDY Oct 10 '19 at 11:40
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@MR.HACKERBUDDY Did you read the first three lines of my answer, by any chance? – 5xum Oct 10 '19 at 11:42
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@MR.HACKERBUDDY The first three lines of my answer tell you that $\pi$ to some power is equal to $q$, where $q$ is any positive ratonal number. I don't know how much more obvious I can make it. – 5xum Oct 10 '19 at 19:22
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I want to say is there any identity of π just as for e {e^(πi)=-1}. – MR. HACKER BUDDY Oct 11 '19 at 00:59