Reducing radicals in the spirit of Euclid

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.