+876 Let $x = 0.\overline{6}$
Thus, $10x = 6.\overline{6}$
Subtract these equations to get: $9x = 6.$
Thus, $x = 0.\overline{6} = \frac{2}{3}.$
We have $3 \sqrt{\frac{2}{3}} = 3 \frac{\sqrt{2}}{\sqrt{3}} = \frac{3 \sqrt{2}}{\sqrt{3}}.$
Then, if we multiply both the numerator and denominator by $\sqrt{3}$ we have
$\frac{3 \sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}} = \frac{3 \sqrt{6}}{3} = \sqrt{6}$.
Thus, the answer is $\sqrt{6}$
