If $\sqrt[n]{x}$ is rational, then so is $x$.
1383
How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?
67605
Different proof that $\sqrt{2}$ is irrational
124448
For every real number $a$ there exists a sequence $r_n$ of rational numbers such that $r_n$ approaches $a$.
75204
Irrationality $\sqrt{2}$ with ideal
148310
Showing rational numbers are algebraic
133496
is $\sqrt{x}$ always positive?
19076
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?
56464
Proving the irrationality of $e^n$
68456
Prove that there exists irrational numbers $x$ and $y$ such that $x + y$ is rational, without using subtraction
147892
Proving if $x<y$ then $\sqrt{x} < \sqrt{y}$
39621
Let a, b, c, d be rational numbers...
156766
Prove that $\sqrt{\frac{n}{n+1}}\notin \mathbb{Q}$
139755
Prove that $\sqrt{2}+\sqrt{3}+\sqrt{5}$ is irrational. Generalise this.
29780
If $x$ is a positive rational but not an integer, is $x^x$ irrational?
92226
Prove $\sqrt6$ is irrational
35837
How to prove that any (integer)$^{1/n}$ that isn't an integer, is irrational?
31436
Show that for all $a\in\mathbb{N}$, there exists $b\in\mathbb{N}$ and square-free integer $c$ such that $\sqrt{a}=b\sqrt{c}$.