\[\frac{3}{7}\sqrt{10\frac{8}{9}} = \frac{3}{7}\sqrt{\frac{98}{9}} = \frac{3}{7}\sqrt{\frac{49 \cdot 2}{9}} =\]
\[= \frac{3}{7} \cdot \frac{7}{3} \cdot \sqrt{2} = \sqrt{2}\]