query: Express $w$ and $\frac1w$ for $w=\frac {\sqrt2+\sqrt3}{\sqrt5-\sqrt3}$ in the simplest form with a rational denominator
positive_passages.docid: 21121
positive_passages.text: Express $w$ and $1/w$ for $w=\frac {\sqrt2+\sqrt3}{\sqrt5-\sqrt3}$ in the simplest form with a rational denominator
negative_passages.docid: 117046
negative_passages.text: Show that $\frac{a^2}{\sqrt{(1+a^3)(1+b^3)}}+\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}}+\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3}$
negative_passages.docid: 129987
negative_passages.text: Find $\lim\frac{1}{\sqrt{n}}\left(\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\ldots+\frac{1}{\sqrt{2n-1}+\sqrt{2n+1}}\right)$
negative_passages.docid: 84335
negative_passages.text: Solving for $\sum_{n = 1}^{\infty} \frac{n^3}{8^n}$?
negative_passages.docid: 107179
negative_passages.text: Partial fraction expansion of $\frac{1}{x(x+1)(x+2)\cdots(x+n)}$
negative_passages.docid: 145632
negative_passages.text: Proving $\frac{1}{\cos^2\alpha}<\frac{\tan\beta-\tan\alpha}{\beta-\alpha}<\frac{1}{\cos^2\beta}$
negative_passages.docid: 128415
negative_passages.text: Prove that $1<\frac{1}{1001}+\frac{1}{1002}+\frac{1}{1003}+\dots+\frac{1}{3001}<\frac43$
negative_passages.docid: 118161
negative_passages.text: Proof of dividing fractional expressions
negative_passages.docid: 136297
negative_passages.text: Evaluate $\lim_{x \to 1} \frac{\sqrt[3]{x} -1}{\sqrt{x} -1}$
negative_passages.docid: 103803
negative_passages.text: Prove from first princibles $\frac{x^2-4}{x-4} \rightarrow -5$ as $x \rightarrow 3$
negative_passages.docid: 148074
negative_passages.text: Analytic expression for the primitive of square root of a quadratic
negative_passages.docid: 116649
negative_passages.text: Value of: $ \frac {1} {\sqrt {4} + \sqrt {5}} + \frac {1} {\sqrt {5} + \sqrt {6}} + ... + \frac {1} {\sqrt {80} + \sqrt {81}}$
negative_passages.docid: 52018
negative_passages.text: Can $\frac{\sqrt{3}}{\sin20^{\circ}}-\frac{1}{\cos20^{\circ}}$ have two values?
negative_passages.docid: 66082
negative_passages.text: Using $\sqrt{1-t}\leq 1-\frac t2$ to show that $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n}\geq\frac1{2\sqrt n}$
negative_passages.docid: 142715
negative_passages.text: Manipulating expression $(\frac{1}{4}+\frac{1}{2^{n+1}})(1-\frac{1}{2^{n+1}})$ in inductive step.
negative_passages.docid: 129880
negative_passages.text: Proving $\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$
negative_passages.docid: 150581
negative_passages.text: How do I show that $\frac a{1 - a^2} + \frac b{1 - b^2} + \frac c{1 - c^2} \ge \frac {3 \sqrt 3}2$