If this expression can be simplified, then its internal part must be equivalent to a square form $$(a+b)^2=a^2 + 2ab+b^2$$ The challenge of this problem is to match each of $a^2$, $b^2$ and $2ab$ with each of $6$, $\sqrt[3]{81}$ and $\sqrt[3]{9}$. $$6+\sqrt[3]{81}+\sqrt[3]{9} = 2\times\sqrt[3]{9}\times\sqrt[3]{3}+(\sqrt[3]{9})^2+(\sqrt[3]{3})^2=(\sqrt[3]{3}+\sqrt[3]{9})^2$$ Therefore, the answer is $\boxed{\sqrt[3]{3}+\sqrt[3]{9}}$.