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As we know that square root of a number $n$ can be found by using a compass and a straight edge, given the line of length $n$.

What I want to know is how to find cube root, fourth root, fifth root or any $n$-th root of a number, by using any tools?

Few days ago I learned that origami (an art of folding paper) can be used as a tool to find cube root and even roots of a cubic equation. So I'm wondering about how to find any other $n$-th roots.

How to decide which kind of tools are necessary for the above mentioned operations? Thanks!

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    $\begingroup$ Fourtn root or any $2^n$th root can be found by repeatedly square-rooting given the number, atleast :) $\endgroup$
    – Sawarnik
    Jan 13 '15 at 16:37
  • $\begingroup$ I've edited your post to include a reference which gives details about how to solve cubic equations using origami. $\endgroup$
    – MvG
    Jan 13 '15 at 20:11
  • $\begingroup$ Archimedes invented an incredibly simple tool for trisecting any angle between $0$ and $\pi$. Trisections are related to cubic equations. $\endgroup$ Mar 7 '19 at 6:43
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Galois theory tells us that the only constructible roots are the $2^n$'th roots. These can be constructed by repeating the square root construction $n$ times. More precisely, if you start with the rational numbers, and allow only straightedge and compass operations on them, you generate the field of constructible numbers, which is characterized by the property of being the smallest field extension of the rationals which is closed under taking square roots. This implies that from a single line segment, the only constructible roots of its length are the $2^n$'th roots.

It is known the origami has enough structure to solve 3rd degree polynomial equations. Another system with the same expressive power can be found in the ancient greek/persian conic geometry. Indeed, roots of cubic polynomials can be found by intersections of conics. See e.g. here.

It is known that some (but perhaps not all) quintic roots admit construction by compass and marked ruler operations, see here, and that all roots which are constructible in this system are of degree 2, 3, 4, 5 or 6.

The general problem of determining what operations one must allow to enable construction of $\sqrt[n]{x}$ is an open problem, it seems. And a hard one at that.

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  • $\begingroup$ You can construct degree 4 roots with a marked ruler and compasses. True, the theory in the referenced paper describes neusis constructive numbers as lying is a "2-3-5-6" tower over the rationale, but the radical solution for fourth-degree equations guarantees that these equations are also covered by the "2-3" part. $\endgroup$ Mar 7 '19 at 1:38
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Its seems that there are many extended constructions for the 3-th (cubic) root of two, for example this one based on J. H. Conway and R. K. Guy, The Book of Numbers, New York: Copernicus Books/Springer, 2006 pp. 194–195.:

Demonstration shows a construction of $\sqrt[3]{2}$ using a ruler with two marks http://demonstrations.wolfram.com/ConstructingTheCubeRootOfTwo/

Using marked ruler is called neusis construction. It is more powerful than a conic drawing tool. But there are also negative results, and it is still open as to whether a regular 25-gon or 31-gon is constructible using this tool.

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