After thinking some time about denesting radicals, I wondered if it was possible to denest a radical in the form $\sqrt[a]{\sqrt[b]{\alpha}+\sqrt[c]{\beta}}$
I thought about rewriting the inside to a power equal to that of $a$. For example: $\sqrt[3]{\sqrt[3]{13}+3\sqrt[3]{4}}=x\sqrt[3]{13}+y\sqrt[3]{4}$. And cubing both sides would give you a Polynomial. Combining terms would give you $x$ and $y$ etc. But I thought that the process would take too long and probably become tedious, especially when $a=10$ or maybe even $50$.
So at this point, I have no idea... anything (including ideas/suggestions) is welcome.