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Let $x^3+\frac{1}{x^3}$ and $x^4+\frac{1}{x^4}$ are rational numbers. Show that $x+\frac{1}{x}$ is rational.

$0.101001000100001000001$... is irrational. [duplicate]

The contradiction method used to prove that the square root of a prime is irrational

rational number solutions to $\frac{a}{a^2+1} + \frac{b}{b^2+1} = \frac{c}{c^2+1}$ with $abc\ne 0$

Proof that $\sqrt6 - \sqrt2 - \sqrt3$ is irrational. [duplicate]

Shorter proof of irrationality of $\sqrt{2}$?

Why aren't all square roots irrational? [duplicate]

Why is $\sum_{i=1}^n a$ always irrational if $n>0$ and $a$ is irrational?

Follow-up Question: Proof of Irrationality of $\sqrt{3}$

Prove that $5^{1/3}+7^{1/2}$ is irrational

Is $x^{1-\frac{1}{n}}+ (1-x)^{1-\frac{1}{n}}$ always irrational if $x$ is rational?

Prove that $\sqrt 5$ is irrational

$\log_2 13$ is irrational

Prove that $\sqrt 2 +\sqrt 3$ is irrational. [duplicate]

Proving $\sqrt2$ is irrational

New posts in rationality-testing

Proving that $\sqrt{2}+\sqrt{3}$ is irrational [duplicate]

If $x$ is a positive rational but not an integer, is $x^x$ irrational?

Prove or disprove that $\sqrt{2+\frac{1}{n}}$ is irrational for $n \in \mathbb{Z}^+$

Irrationality of $\pi^2$ and $\pi^3$

Show that $({\sqrt{2}\!+\!1})^{1/n} \!+ ({\sqrt{2}\!-\!1})^{1/n}\!\not\in\mathbb Q$

Let $x^3+\frac{1}{x^3}$ and $x^4+\frac{1}{x^4}$ are rational numbers. Show that $x+\frac{1}{x}$ is rational.

$0.101001000100001000001$... is irrational. [duplicate]

The contradiction method used to prove that the square root of a prime is irrational

rational number solutions to $\frac{a}{a^2+1} + \frac{b}{b^2+1} = \frac{c}{c^2+1}$ with $abc\ne 0$

Proof that $\sqrt6 - \sqrt2 - \sqrt3$ is irrational. [duplicate]

Shorter proof of irrationality of $\sqrt{2}$?

Why aren't all square roots irrational? [duplicate]

Why is $\sum_{i=1}^n a$ always irrational if $n>0$ and $a$ is irrational?

Follow-up Question: Proof of Irrationality of $\sqrt{3}$

Prove that $5^{1/3}+7^{1/2}$ is irrational

Is $x^{1-\frac{1}{n}}+ (1-x)^{1-\frac{1}{n}}$ always irrational if $x$ is rational?

Prove that $\sqrt 5$ is irrational

$\log_2 13$ is irrational

Prove that $\sqrt 2 +\sqrt 3$ is irrational. [duplicate]

Proving $\sqrt2$ is irrational