We have
$$\sqrt[3]{ \frac{1 - 3 \sqrt[3]{3}}{11- 3 \sqrt[3]{3}}} = \frac{1 - \sqrt[3]{3}}{2- \sqrt[3]{3}}$$
since
$$\require{cancel} \sqrt[3]{ \frac{\ 1 - 3^{1/3} 3}{11 - 3^{1/3} 3} }=\frac{(\ 1 - 3^{1/3}\ 3)^{1/3} }{(11 - 3^{1/3}\ 3)^{1/3}}= \frac{(\ 1 - 3^{1/3}\cancel{3})^{1/\cancel{3}} }{(\textrm{II} - 3^{1/3}\bcancel{3})^{1/\bcancel{3}}}= \frac{1 - 3^{1/3}}{2 - 3^{1/3}} $$
I was told that the above proof is wrong. However, it does check numerically, both sides equal $-0.7929165927333\ldots$. Where is my error?