query: Show that if $n > 1$ is not a $k$-th power, then its $k$-th root is irrational, i.e. not of the form $u/v$ for $u, v ∈ N$.
positive_passages.docid: 1383
positive_passages.text: How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?
negative_passages.docid: 11710
negative_passages.text: How to show that $\mathbb{Q}$ is not $G_\delta$
negative_passages.docid: 30622
negative_passages.text: Proving $\sqrt 3$ is irrational.
negative_passages.docid: 80304
negative_passages.text: Roots of $\,ax^2+bx+c\,$ are irrational if $a,b,c$ are odd integers
negative_passages.docid: 122226
negative_passages.text: Incorrect logic in popular proof of the irrationality of $\sqrt2$?
negative_passages.docid: 151439
negative_passages.text: $\sqrt{1-x^2}$ is not a rational function
negative_passages.docid: 121076
negative_passages.text: Show $x^p-t$ has no root in the field $\mathbb{F}_p(t)$
negative_passages.docid: 131634
negative_passages.text: Prove the irrationality of $0.235711131719...$
negative_passages.docid: 93029
negative_passages.text: Prove that when $d$ is in $\mathbb{Q} \setminus\mathbb{Z}$, $\mathbb{Z}[\sqrt{d}]$ is not a finitely generated abelian group.
negative_passages.docid: 56464
negative_passages.text: Proving the irrationality of $e^n$
negative_passages.docid: 87735
negative_passages.text: Prove that the roots $\in \Bbb R$ of $x^3+x+1=0$ aren't rational without RRT
negative_passages.docid: 34491
negative_passages.text: Is there a way to show that $\sqrt{p_{n}} < n$?
negative_passages.docid: 75639
negative_passages.text: Showing that $\sqrt[3]{2}\notin\Bbb Q(\alpha_1,...,\alpha_k)$ where $\alpha_i^2\in\Bbb Q\ \forall i$
negative_passages.docid: 18360
negative_passages.text: Proof that the irrational numbers are uncountable
negative_passages.docid: 91101
negative_passages.text: Proving $\sqrt2$ is irrational
negative_passages.docid: 150088
negative_passages.text: Prove that $\sqrt {n − 1} +\sqrt {n + 1}$ is irrational for every integer $n ≥ 1$
negative_passages.docid: 63409
negative_passages.text: If $x^2+ax+b=0$ has a rational root, show that it is in fact an integer