Denesting radicals $\sqrt[3]{-22+15\sqrt[3]{3}+9\sqrt[3]{9}}$ and $\sqrt[3]{8-9\sqrt[3]{3}+3\sqrt[3]{9}}$ I am trying to do denesting radicals:$$\sqrt[3]{-22+15\sqrt[3]{3}+9\sqrt[3]{9}}$$ and $$\sqrt[3]{8-9\sqrt[3]{3}+3\sqrt[3]{9}}$$
I tried to find Ramanujan polynomial like this link denesting radicals
But it doesn't work. I also tried to solve the system of equation $\sqrt[3]{-22+15\sqrt[3]{3}+9\sqrt[3]{9}}=a+b\sqrt[3]{3}+c\sqrt[3]{9}$ but it led to a scary-looking one.

Using Wolfram Alpha:
WA:
$\alpha=-22+15\sqrt[3]{3}+9\sqrt[3]{9}$ is a root of $f(x)=x^3 + 66 x^2 + 237 x - 32768$.
WA:
$f(x^3)=(x^3 - 6 x^2 + 21 x - 32) (x^6 + 6 x^5 + 15 x^4 + 62 x^3 + 249 x^2 + 672 x + 1024)$.
This suggests that the (real) cubic root of $\alpha$ is a root of $g(x)=x^3 - 6 x^2 + 21 x - 32$, because the other factor is irreducible and has degree 6 and we expect the answer to be in the cubic field $\mathbb Q(\sqrt[3]3)$.
WA:
the only real root of $g$ is $\beta= 2 - \sqrt[3]{3} + \sqrt[3]{9}$.
WA:
it easy to check that $\beta^3=\alpha$.
