Решение:

1. $$3 \frac{3}{4} \cdot 1 \frac{7}{9} = \frac{15}{4} \cdot \frac{16}{9} = \frac{15 \cdot 16}{4 \cdot 9} = \frac{5 \cdot 4}{1 \cdot 3} = \frac{20}{3} = 6 \frac{2}{3}$$
2. $$3 \frac{2}{4} : 2 \frac{2}{9} = 3 \frac{1}{2} : 2 \frac{2}{9} = \frac{7}{2} : \frac{20}{9} = \frac{7}{2} \cdot \frac{9}{20} = \frac{7 \cdot 9}{2 \cdot 20} = \frac{63}{40} = 1 \frac{23}{40}$$
3. $$7 \frac{7}{9} \cdot 5 \frac{2}{5} \cdot 1 \frac{1}{14} = \frac{70}{9} \cdot \frac{27}{5} \cdot \frac{15}{14} = \frac{70 \cdot 27 \cdot 15}{9 \cdot 5 \cdot 14} = \frac{10 \cdot 3 \cdot 3}{1 \cdot 1 \cdot 2} = \frac{90}{2} = 45$$

Ответ: 1) $$6 \frac{2}{3}$$; 2) $$1 \frac{23}{40}$$; 3) $$45$$