Cardan Polynomials and the Reduction of Radicals

@article{Osler2001CardanPA,
  title={Cardan Polynomials and the Reduction of Radicals},
  author={T. Osler},
  journal={Mathematics Magazine},
  year={2001},
  volume={74},
  pages={26 - 32}
}
  • T. Osler
  • Published 2001
  • Mathematics
  • Mathematics Magazine
A quick check with a calculator shows that this expression is really 1, but how can one algebraically manipulate it into that result? All attempts to do so led the author back to the original cubic from which it started. (The reader might try now to reduce (3) to the value 1.) That such a combination of square and cube roots should produce a rational number, let alone an integer, is startling. Later we will see that the individual radicals are related to the golden section and its reciprocal: 
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