  Let $p$ be an odd natural number $\ge 3$. Inspired by results from Euclid's
{\em Elements}, we express the irrational $$y=\sqrt[p]{d+\sqrt R}, $$ whose
degree is $2p$, as a polynomial function of irrationals of degrees $\le p$. In
certain cases $y$ is expressed by simple radicals. This reduction of the degree
exhibits remarkably regular patterns of the polynomials involved. The proof is
based on hypergeometric summation, in particular, on Zeilberger's algorithm.
