State whether \((\pi)^e\) is a rational number or an irrational number with proof.
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2 years, 7 months ago
It's transcendal number.
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Proving that a number is an irrational is extremely difficult. As Shivang Jindal has mentioned, there a lot of "at least one of them is irrational" cases, but to prove which one is in the stars with respect to difficulty. Even the irrationality of \(\pi\) is very difficult to prove.
Even proving \[ e+\pi , e\pi \] is irrational/rational is unsolved,despite the fact that, at-least one of them has to be irrational![You can show this easily] .
More strongly, at least one of \(e+\pi, e\pi\) is transcendental. Proof here: \(x^2-(e+\pi)x+e\pi\) has roots \(e,\pi\). Assume for contradiction both \(e+\pi, e\pi\) are algebraic. Then \(e,\pi\) are algebraic (since they're roots of a polynomial with algebraic coefficients), contradiction, since \(e,\pi\) are transcendental.