So so long

Let the LHS and RHS be $f_1(x)$ and $f_2(x)$ respectively. Then,

$\begin{aligned} f_1(x) & = x^9 + 18x^6 + 108x^3 + 222 \\ \frac {df_1}{dx} & = 9x^8 + 108x^5 + 324x^2 = 9x^2(x^3+6)^2 \ge 0 \ \forall x \end{aligned}$

Therefore, the LHS $f_1(x)$ is a non-decreasing function for all $x$ .

Similarly, $f_2(x) = \sqrt[3]{x-6}$ $\implies \dfrac {df_2}{dx} = \dfrac 1{(\sqrt[3]{x-6})^2} \ge 0 \ \forall x$ . Again, $f_2(x)$ is a non-decreasing function for all $x$ .

Hence, there is only $\boxed{1}$ solution for $f_1(x) = f_2(x)$ and the solution is $x=-2$ .