On a straight line with marked origin 0 and unit 1, two points x and y are given. Is it possible, by any finite method, to geometrically define x^y if the given y is not a rational but an arbitrary point?
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So, is the question something like: Given three marks $x$, $y$ and $z$ where someone claims that $x^y=z$, is there a finite geometric construction (possibly involving other guessed points) they can use to prove they are right? – hmakholm left over Monica Jun 01 '16 at 12:59
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Answer to an earlier version of the question which asked to construct $x^y$ instead of merely defining it:
No, at least not if "any finite method" means compass and straightedge -- for example this is famously impossible when $x=2$ and $y=1/3$.
hmakholm left over Monica
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not necessarily straightedge and compass. If y is rational, x^y can be defined by a procedure involving similar triangles. The question is what happens if y is irrational. – exp8j Jun 01 '16 at 12:43
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2@J.Avaris You should specify the rules of the game if you want us to play along! – Lynn Jun 01 '16 at 12:44
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Wow I found this construction of $\sqrt[3]{2}$ ... is there something wrong with it? Or does it use more than a compass and straightedge? – Zubin Mukerjee Jun 01 '16 at 12:45
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@J.Avaris: In that case you need to define "construct" and "any finite method" explicitly in the question -- otherwise any attempt to answer will be at risk of having the goalposts moved. (And just because you can define what a cube root is doesn't mean you can necessarily construct it). – hmakholm left over Monica Jun 01 '16 at 12:45
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1@ZubinMukerjee: The figure cannot be constructed just with compass and straightedge -- after drawing AB and the lines from B towards C and D, you need a "sliding ruler with marks" in order to find the line ACD. – hmakholm left over Monica Jun 01 '16 at 12:48
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@HenningMakholm Ok, thanks! Is there some story about $\sqrt[3]{2}$ that made it famous? – Zubin Mukerjee Jun 01 '16 at 12:49
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1@ZubinMukerjee: Doubling the cube is one of three famous ancient geometric construction problems that were only proved impossible (by compass and straightedge) in the 19th century. The other two ones are squaring the circle and trisecting an arbitrary angle. – hmakholm left over Monica Jun 01 '16 at 12:51
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