query: How can I prove that an irrational number can be raised to another one and the result is rational?
positive_passages.docid: 1877
positive_passages.text: Can an irrational number raised to an irrational power be rational?
negative_passages.docid: 35837
negative_passages.text: How to prove that any (integer)$^{1/n}$ that isn't an integer, is irrational?
negative_passages.docid: 19076
negative_passages.text: If a.b.c,d are rationals and x is irrational number, then prove that $\frac{ax+b}{cx+d}$ is usually a irrational number. When do exceptions occur?
negative_passages.docid: 132115
negative_passages.text: Verify my proof of the irrationality of $\sqrt{6}$.
negative_passages.docid: 117366
negative_passages.text: Is there a simple, constructive, 1-1 mapping between the reals and the irrationals?
negative_passages.docid: 96204
negative_passages.text: Irrational number $\sqrt2$ proof
negative_passages.docid: 22408
negative_passages.text: Jutifying: irrational + rational = irrational
negative_passages.docid: 111094
negative_passages.text: Help with proving that $\pi$ is irrational
negative_passages.docid: 13592
negative_passages.text: How can $\pi$ be irrational if it can be calculated using an infinite series of rational numbers?
negative_passages.docid: 19676
negative_passages.text: Is it known whether ${\sqrt{2}}^{\sqrt{2}}$ is irrational?
negative_passages.docid: 101028
negative_passages.text: Irrational numbers and Borel Sets
negative_passages.docid: 64885
negative_passages.text: Irrational and rational sequence proof
negative_passages.docid: 91101
negative_passages.text: Proving $\sqrt2$ is irrational
negative_passages.docid: 4946
negative_passages.text: Non-existence of irrational numbers?
negative_passages.docid: 11069
negative_passages.text: Deciding whether $2^{\sqrt2}$ is irrational/transcendental
negative_passages.docid: 67031
negative_passages.text: use contradiction to prove that the square root of $p$ is irrational
negative_passages.docid: 41767
negative_passages.text: Sequence of rational numbers converging to an arbitrary irrational number $\beta$.