\[\sqrt[3]{- 4\frac{17}{27}} + 6\sqrt[4]{3\frac{13}{81}} = \sqrt[3]{- \frac{125}{27}} + 6\sqrt[4]{\frac{256}{81}} =\]
\[= - \frac{5}{3} + 6 \cdot \frac{4}{3} = - 1\frac{2}{3} + 8 = 6\frac{1}{3}.\]