Quintic Expression

Let $a = u + v$ where $u = (41 + 29\sqrt{2})^{1/5}$ and $v = (41 -29\sqrt{2})^{1/5}$ Then $uv = (41 + 29\sqrt{2})^{1/5}(41 - 29\sqrt{2})^{1/5} = (41^2 - 2*29^2)^{1/5} = (-1)^{1/5} = -1$ Let $a = (u + v)$ $\begin{aligned} a^5 & = u^5 + 5u^4v + 10u^3v^2 + 10u^2v^3 + 5uv^4 +v^5 \\ & = u^5 +v^5 + 5uv(u^3 + v^3) + 10u^2v^2(u + v) \\ & = u^5 +v^5 + 5uv[(u + v)^3 - 3uv(u + v)] + 10(uv)^2(u + v) \\ & = 41 + 29\sqrt{2} + 41 - 29\sqrt{2} + 5(-1)[a^3 - 3(-1)a] + 10(-1)^2a \\ & = 82 - 5a^3 - 15a +10a \end{aligned}$

Giving $a^5 + 5a^3 + 5a - 82 =0$ ; Factoring : $(a - 2)(a^4 +2a^3 + 9a^2 + 18a + 41) = 0$ Which implies that $a = \boxed{2}$