Simplify $\sqrt[3]{9\sqrt3-11\sqrt2}$ Simplify $$\sqrt[3]{9\sqrt3-11\sqrt2}$$
How can we actually simplify this radical?

Apply the denesting formula
$$\sqrt[3]{a\sqrt d+b \sqrt c}=\frac12\sqrt[3]{\frac{3bt-a}d}\left(\sqrt d+\frac1t \sqrt c\right)$$
with $t$ satisfying $t^3-\frac{3a}bt^2+\frac{3c}d t-\frac{ac}{bd}=0$. Then
$$\sqrt[3]{9\sqrt3-11\sqrt2} = \frac12\sqrt[3]{-11t-3}\left(\sqrt 3+\frac1t \sqrt 2\right) $$
where $t^3+\frac{27}{11}t^2+2t+\frac{6}{11}=0$, yielding $t=-1$. Plug into above expression to obtain
$$\sqrt[3]{9\sqrt3-11\sqrt2}
=\sqrt{3}-\sqrt2$$
