Conjugates of $12^{1/5}+54^{1/5}-144^{1/5}+648^{1/5}$ over $\mathbb{Q}$ After much manual computation I found the minimal polynomial to be $x^5+330x-4170$, although I would very much like to know if there's a clever way to see this. I suspect there is from seeing that the prime factorisations of the four integers are as various powers of $2$ and $3$ and because the number can be written as: $$12^{1/5}+54^{1/5}-(12^{1/5})^2+(12^{1/5}\cdot 54^{1/5})$$
But I haven't yet been able to find anything better than manual computation of the fifth power.
However, from here I am lost, I'm not sure how to solve this equation. A quick internet search returns much information on the general case in terms of the "Bring-Gerrard normal form" but the book from which this problem was taken hasn't gone into any detail on general methods for solving polynomials, so I am trying to find a solution that doesn't require any heavy machinery.

Noting that $12=\dfrac{18^5}{54^3}$ and $144 = \dfrac{6^5}{54}$, letting $\alpha = 54^{1/5}$, we see that the above is:
$$18\alpha^{-3} + \alpha - 6\alpha^{-1} + 18\alpha^{-2} = \frac{18\alpha^2 + 54\alpha - 6\alpha^4 + 18\alpha^3}{54}$$
Now replace $\alpha$ by conjugates of $\alpha$.
