query: Show that $\sqrt{6+4\sqrt{2}}-\sqrt{2}$ is rational using the rational zeros theorem
positive_passages.docid: 153043
positive_passages.text: Show that $\sqrt{4+2\sqrt{3}}-\sqrt{3}$ is rational using the rational zeros theorem
negative_passages.docid: 125702
negative_passages.text: There is a square $Q$ consisting of $(0,0), (2,0), (0,2), (2,2)$
negative_passages.docid: 3390
negative_passages.text: finding the real values of $x$ such that : $x=\sqrt{2+\sqrt{2-\sqrt{2+x}}}$
negative_passages.docid: 88425
negative_passages.text: Prove that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$
negative_passages.docid: 22224
negative_passages.text: Define $S\equiv\{ x\in \mathbb{Q}\mid x^2<2\}$. Show that $\sup S=\sqrt{2} $.
negative_passages.docid: 133164
negative_passages.text: Rational solutions for $x^2+y^2=3$
negative_passages.docid: 81921
negative_passages.text: If $f(x + y) = f(x) + f(y)$ showing that $f(cx) = cf(x)$ holds for rational $c$
negative_passages.docid: 88429
negative_passages.text: Prove $\sqrt{2} + \sqrt{5}$ is irrational
negative_passages.docid: 141184
negative_passages.text: Proof that a degree 4 polynomial has at least two roots
negative_passages.docid: 22800
negative_passages.text: Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares?
negative_passages.docid: 28347
negative_passages.text: Prove that $\sqrt 2 + \sqrt 3$ is irrational
negative_passages.docid: 80002
negative_passages.text: zeros of a polynomial
negative_passages.docid: 158933
negative_passages.text: Rational solutions of $y^2 = x^3 - x$
negative_passages.docid: 154322
negative_passages.text: Localizing the zeros of $z^4+z+1$
negative_passages.docid: 6582
negative_passages.text: Prove the existence of the square root of $2$.
negative_passages.docid: 122083
negative_passages.text: Prove that $\sqrt[3]{2} +\sqrt{5}$ is algebraic over $\mathbb Q$
negative_passages.docid: 48035
negative_passages.text: Prove that square root of 2 is irrational using the principle of Mathematical Induction