Genady Posted April 26, 2023 Share Posted April 26, 2023 Link to comment Share on other sites More sharing options...
Genady Posted April 29, 2023 Author Share Posted April 29, 2023 Hint: if you think to take the cube of this expression, you are on the right track. Link to comment Share on other sites More sharing options...
Genady Posted May 1, 2023 Author Share Posted May 1, 2023 A little algebra help for those who don't remember the formula: taking the cube of the expression in OP gives at first, Spoiler Now, to simplify... Link to comment Share on other sites More sharing options...
Genady Posted May 3, 2023 Author Share Posted May 3, 2023 (edited) So, after taking cube of the expression in question, and simplifying, we get it (i.e., the cube) being equal to Spoiler Do you see something peculiar here? Edited May 3, 2023 by Genady Link to comment Share on other sites More sharing options...
Sensei Posted May 3, 2023 Share Posted May 3, 2023 (edited) Such math puzzles make little sense in the age of Wolfram Alpha. It solves them right away. An opportunity to try handmade Latex: [math]\sqrt[3]{7+\sqrt{50}}+\sqrt[3]{7-\sqrt{50}}[/math] [math]\sqrt[3]{7+\sqrt{50}}+\sqrt[3]{7-\sqrt{50}}[/math] Edited May 3, 2023 by Sensei Link to comment Share on other sites More sharing options...
Genady Posted May 3, 2023 Author Share Posted May 3, 2023 9 minutes ago, Sensei said: Such math puzzles make little sense in the age of Wolfram Alpha. It solves them right away. An opportunity to try handmade Latex: 7+50−−√−−−−−−−√3+7−50−−√−−−−−−−√3 [math]\sqrt[3]{7+\sqrt{50}}+\sqrt[3]{7-\sqrt{50}}[/math] What do you mean, it solves them right away? Link to comment Share on other sites More sharing options...
Sensei Posted May 3, 2023 Share Posted May 3, 2023 18 minutes ago, Genady said: What do you mean, it solves them right away? https://www.wolframalpha.com/input?i=cube+root+of(+7%2Bsqrt(50))%2Bcube+root+of(+7-sqrt(50)) Link to comment Share on other sites More sharing options...
Genady Posted May 3, 2023 Author Share Posted May 3, 2023 12 minutes ago, Sensei said: https://www.wolframalpha.com/input?i=cube+root+of(+7%2Bsqrt(50))%2Bcube+root+of(+7-sqrt(50)) It doesn't let me see the whole step-by-step solution, but it looks like it just keeps simplifying the original expression. If so, it misses the beautiful insight: after the first step of taking the cube, the puzzle boils down to the simple equation, Spoiler x3 = 14 - 3x which is quickly solved by inspection. Link to comment Share on other sites More sharing options...
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