query: Is $ (((\sqrt{2})^ \sqrt{2})^ \sqrt{2})^{\cdots} $ an irrational number?
positive_passages.docid: 20532
positive_passages.text: Are these solutions of $2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$ correct?
negative_passages.docid: 10882
negative_passages.text: irrationality of $\sqrt{2}^{\sqrt{2}}$.
negative_passages.docid: 91101
negative_passages.text: Proving $\sqrt2$ is irrational
negative_passages.docid: 44630
negative_passages.text: Is $\frac{1}{\sqrt{2}}$ a fraction or not?
negative_passages.docid: 107142
negative_passages.text: If $(x_n) \to x$ then $(\sqrt[n]{x_1x_2\cdots x_n}) \to x$
negative_passages.docid: 158368
negative_passages.text: Irrationals: A Group?
negative_passages.docid: 88429
negative_passages.text: Prove $\sqrt{2} + \sqrt{5}$ is irrational
negative_passages.docid: 18360
negative_passages.text: Proof that the irrational numbers are uncountable
negative_passages.docid: 90276
negative_passages.text: Is $\sqrt{2}\in{\Bbb Z}[\sqrt{2}+\sqrt{3}]$ true?
negative_passages.docid: 70315
negative_passages.text: Process to show that $\sqrt 2+\sqrt[3] 3$ is irrational
negative_passages.docid: 127830
negative_passages.text: proof that $\sqrt(2)$ is irrational / why is the fraction irreducible
negative_passages.docid: 67031
negative_passages.text: use contradiction to prove that the square root of $p$ is irrational
negative_passages.docid: 118374
negative_passages.text: Is there a proof that $\pi$ is an irrational number?
negative_passages.docid: 53945
negative_passages.text: The square root of a prime number is irrational
negative_passages.docid: 23752
negative_passages.text: Proving Irrationality
negative_passages.docid: 134377
negative_passages.text: The minimum number of digits after the floating-point, which uniquely identify every irrational square root
negative_passages.docid: 8589
negative_passages.text: $\sqrt[b_1]{a_1}+\sqrt[b_2]{a_2}+\cdots+\sqrt[b_k]{a_k}$ not an integer