<s>[INST] 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.
 [/INST] (Too long for a comment.)

this is the system of equations after raise power of 3
$$
\begin{align}
p(a,b,c) = a^3+3b^3+9c^3+18abc+22 &= 0
\\ q(a,b,c) = a^2b+3ac^2+3b^2c-5 &= 0
\\ r(a,b,c) = a^2c+ab^2+3bc^2-3 &= 0
\end{align}
$$

Not something you'd do by hand, but that's an algebraic system which can always be reduced to one single polynomial equation using resultants, and then checked for rational solutions using the rational root theorem.
In this case for example, with WA's assistance you get 1 & 2
$\to$ 3 with the only rational root $c=1\,$:
$$
\begin{align}
u(b,c) = \text{res}_a (p,q) &= -3 b^9 + 27 b^6 c^3 - 22 b^6 - 81 b^5 c + 648 b^3 c^6 - 66 b^3 c^3
\\ &\quad - 486 b^2 c^4 - 198 b^2 c - 243 b c^2 + 81 c^9 + 396 c^6 + 484 c^3 - 27
\\v(b,c) = \text{res}_a(q,r) &= 3 b^7 c - 5 b^5 - 18 b^4 c^4 + 9 b^3 c^2 + 15 b^2 c^3 + 9 b^2 + 27 b c^7
\\ &\quad - 30 b c - 27 c^5 + 25 c^2
\\ w(c) = \text{res}_b(u,v) &= -9 (c - 1) (c^2 + c + 1) (3 c^3 + 1) (9 c^3 - 8) (\small{\dots\text{<degree 63 factor>} \dots})
\end{align}
$$
</s>