Prove that there is an irrational number $a$ such that $a^{\sqrt3}$ is rational.
1877
Can an irrational number raised to an irrational power be rational?
10882
irrationality of $\sqrt{2}^{\sqrt{2}}$.
110282
Use the unique factorization for integers theorem and the definition of logarithm to prove that $\log_3 (7)$ is irrational.
126074
Proof of the irrationality of $\sqrt{3}$ - logic question
144244
Bound on difference between irrational square root and rational number
139755
Prove that $\sqrt{2}+\sqrt{3}+\sqrt{5}$ is irrational. Generalise this.
56877
Show that $\sqrt{4 + 2\sqrt{3}} - \sqrt{3}$ is rational.
126199
Proof of $\pi$ not being a quadratic irrational number.
53945
The square root of a prime number is irrational
35837
How to prove that any (integer)$^{1/n}$ that isn't an integer, is irrational?
43985
Why must $a$ and $b$ both be coprime when proving that the square root of two is irrational?
70315
Process to show that $\sqrt 2+\sqrt[3] 3$ is irrational
44600
How can I prove: if $p $ is prime and $n>1$, then $ p^{\frac1n} $ is irrational?
40009
Show that $e^{\sqrt 2}$ is irrational
75204
Irrationality $\sqrt{2}$ with ideal
151153
Prove $\frac{a(a^2+2)}{3}$ is an integer for all integer $a\geqslant 1$
26109
Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$