Could $\pi$, raised to some power can give rational result? [duplicate]

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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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1 Answer

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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.

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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.

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* Proving $\pi$ is transcendental was quite a feat back when it was first done and was by no means an easy task!

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