[Dr. Vogler's trick for removing radicals from equations][1], which essentially amounts to turning a linear equation into a system of equations for a specific set of monomials, each of which is considered a different variable.
Of all the monomial-variables that have been determined, all that do not represent a radical of the original equation, are discarded.

also related are these questions and answers:

- https://mathoverflow.net/questions/177765/rewrite-sum-of-radicals-equation-as-polynomial-equation

- https://mathoverflow.net/questions/176610/tools-for-removing-radicals-from-equations

 [1]: https://web.archive.org/web/20180521122321/http://mathforum.org/library/drmath/view/70442.html