1) $$6\frac{6}{7} \cdot \frac{3}{8} = \frac{6 \cdot 7 + 6}{7} \cdot \frac{3}{8} = \frac{48}{7} \cdot \frac{3}{8} = \frac{6}{7} \cdot \frac{3}{1} = \frac{18}{7} = 2\frac{4}{7}$$
2) $$5\frac{1}{7} \cdot 1\frac{8}{27} = \frac{5 \cdot 7 + 1}{7} \cdot \frac{1 \cdot 27 + 8}{27} = \frac{36}{7} \cdot \frac{35}{27} = \frac{4}{1} \cdot \frac{5}{3} = \frac{20}{3} = 6\frac{2}{3}$$
3) $$2\frac{5}{8} \cdot 2\frac{2}{7} = \frac{2 \cdot 8 + 5}{8} \cdot \frac{2 \cdot 7 + 2}{7} = \frac{21}{8} \cdot \frac{16}{7} = \frac{3}{1} \cdot \frac{2}{1} = 6$$
4) $$1\frac{8}{9} \cdot 1\frac{1}{8} \cdot 2\frac{2}{3} = \frac{1 \cdot 9 + 8}{9} \cdot \frac{1 \cdot 8 + 1}{8} \cdot \frac{2 \cdot 3 + 2}{3} = \frac{17}{9} \cdot \frac{9}{8} \cdot \frac{8}{3} = \frac{17}{1} \cdot \frac{1}{1} \cdot \frac{1}{3} = \frac{17}{3} = 5\frac{2}{3}$$
Ответ: 1) $$2\frac{4}{7}$$; 2) $$6\frac{2}{3}$$; 3) $$6$$; 4) $$5\frac{2}{3}$$