352. Найдите значение выражения: 1) $$4 \frac{2}{3} \cdot 6 - 1 \frac{23}{42} + 2 \frac{1}{8} \cdot \frac{29}{51}$$; 2) $$(5 \frac{3}{10} - 3 \frac{4}{5}) \cdot \frac{5}{38} + 2 \frac{1}{84} - \frac{1}{12} \cdot \frac{4}{15}$$; 3) $$(3 \frac{1}{3})^{2} - 2 \frac{13}{16} \cdot 2 \frac{2}{5}$$; 4) $$( \frac{5}{12} + \frac{13}{20})^{2} \cdot \frac{13}{32}$$

Решение:

1. $$4 \frac{2}{3} \cdot 6 - 1 \frac{23}{42} + 2 \frac{1}{8} \cdot \frac{29}{51} = \frac{14}{3} \cdot 6 - \frac{65}{42} + \frac{17}{8} \cdot \frac{29}{51} = 28 - \frac{65}{42} + \frac{17 \cdot 29}{8 \cdot 51} = 28 - \frac{65}{42} + \frac{493}{408} = \frac{28 \cdot 42 \cdot 408 - 65 \cdot 408 + 493 \cdot 42}{42 \cdot 408} = \frac{479088 - 26520 + 20706}{17136} = \frac{473274}{17136} = \frac{236637}{8568} = 27 \frac{5181}{8568} = 27 \frac{1727}{2856}$$
2. $$(5 \frac{3}{10} - 3 \frac{4}{5}) \cdot \frac{5}{38} + 2 \frac{1}{84} - \frac{1}{12} \cdot \frac{4}{15} = (\frac{53}{10} - \frac{19}{5}) \cdot \frac{5}{38} + \frac{169}{84} - \frac{4}{12 \cdot 15} = (\frac{53 - 19 \cdot 2}{10}) \cdot \frac{5}{38} + \frac{169}{84} - \frac{1}{45} = (\frac{53 - 38}{10}) \cdot \frac{5}{38} + \frac{169}{84} - \frac{1}{45} = \frac{15}{10} \cdot \frac{5}{38} + \frac{169}{84} - \frac{1}{45} = \frac{3}{2} \cdot \frac{5}{38} + \frac{169}{84} - \frac{1}{45} = \frac{15}{76} + \frac{169}{84} - \frac{1}{45} = \frac{15 \cdot 84 \cdot 45 + 169 \cdot 76 \cdot 45 - 1 \cdot 76 \cdot 84}{76 \cdot 84 \cdot 45} = \frac{56700 + 576180 - 6384}{287280} = \frac{626596}{287280} = \frac{156649}{71820} = 2 \frac{129}{71820}$$
3. $$(3 \frac{1}{3})^{2} - 2 \frac{13}{16} \cdot 2 \frac{2}{5} = (\frac{10}{3})^{2} - \frac{45}{16} \cdot \frac{12}{5} = \frac{100}{9} - \frac{45 \cdot 12}{16 \cdot 5} = \frac{100}{9} - \frac{9 \cdot 3}{4} = \frac{100}{9} - \frac{27}{4} = \frac{100 \cdot 4 - 27 \cdot 9}{36} = \frac{400 - 243}{36} = \frac{157}{36} = 4 \frac{13}{36}$$
4. $$( \frac{5}{12} + \frac{13}{20})^{2} \cdot \frac{13}{32} = (\frac{5 \cdot 5 + 13 \cdot 3}{60})^{2} \cdot \frac{13}{32} = (\frac{25 + 39}{60})^{2} \cdot \frac{13}{32} = (\frac{64}{60})^{2} \cdot \frac{13}{32} = (\frac{16}{15})^{2} \cdot \frac{13}{32} = \frac{256}{225} \cdot \frac{13}{32} = \frac{256 \cdot 13}{225 \cdot 32} = \frac{8 \cdot 13}{225} = \frac{104}{225}$$