query: Prove that $\sqrt6 - \sqrt2 - \sqrt3$ is irrational.
positive_passages.docid: 21237
positive_passages.text: $\sqrt{6}-\sqrt{2}-\sqrt{3}$ is irrational
negative_passages.docid: 89716
negative_passages.text: Is there a proof of the irrationality of $\sqrt{2}$ that involves modular arithmetic?
negative_passages.docid: 99311
negative_passages.text: Prove $2^{1/3} + 2^{2/3}$ is irrational
negative_passages.docid: 2926
negative_passages.text: The contradiction method used to prove that the square root of a prime is irrational
negative_passages.docid: 129433
negative_passages.text: $\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$ is Irrational
negative_passages.docid: 20315
negative_passages.text: $\sqrt[31]{12} +\sqrt[12]{31}$ is irrational
negative_passages.docid: 153043
negative_passages.text: Show that $\sqrt{4+2\sqrt{3}}-\sqrt{3}$ is rational using the rational zeros theorem
negative_passages.docid: 129171
negative_passages.text: Prove that $1+\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6 \cdot 9}+.........=\sqrt{3}$
negative_passages.docid: 144244
negative_passages.text: Bound on difference between irrational square root and rational number
negative_passages.docid: 88425
negative_passages.text: Prove that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$
negative_passages.docid: 147777
negative_passages.text: If $P$ and $Q$ are distinct primes, how to prove that $\sqrt{PQ}$ is irrational?
negative_passages.docid: 126074
negative_passages.text: Proof of the irrationality of $\sqrt{3}$ - logic question
negative_passages.docid: 17930
negative_passages.text: Please explain this step in proving the square root of 3 is irrational
negative_passages.docid: 127830
negative_passages.text: proof that $\sqrt(2)$ is irrational / why is the fraction irreducible
negative_passages.docid: 1190
negative_passages.text: Square Root of Rational Number $\frac{A}{B}$
negative_passages.docid: 56877
negative_passages.text: Show that $\sqrt{4 + 2\sqrt{3}} - \sqrt{3}$ is rational.
negative_passages.docid: 59391
negative_passages.text: $p,q,r$ primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational.