105. Найдите значение выражения: а) \( \frac{17}{25} \cdot 2 \frac{1}{7} - 2 \frac{4}{7} \cdot 1 \frac{2}{5} \); б) \( \frac{1}{13} \cdot \left( 2 \frac{3}{8} - 1 \frac{5}{6} \right) \cdot 2 \frac{2}{5} + \frac{9}{10} \); в) \( 4 \frac{3}{4} - 1 \frac{2}{5} \cdot 2 \frac{1}{2} \right) \cdot 1 \frac{3}{5} \); г) \( 1 \frac{1}{2} \cdot \frac{2}{9} - \left( 6 - 5 \frac{3}{5} \right)^2 \); д) \( 4 \frac{1}{15} - 3 \frac{9}{10} \right) \cdot 6 \frac{6}{7} + 2 \); е) \( \frac{1}{3} \cdot \left( 4 - \left( 1 \frac{1}{2} \right)^3 \right) \cdot \left( 3 - 2 \frac{7}{15} \right) \)

Решение:

1. а) \( \frac{17}{25} \cdot 2 \frac{1}{7} - 2 \frac{4}{7} \cdot 1 \frac{2}{5} = \frac{17}{25} \cdot \frac{15}{7} - \frac{18}{7} \cdot \frac{7}{5} = \frac{17 \cdot 3}{5 \cdot 7} - \frac{18 \cdot 1}{1 \cdot 5} = \frac{51}{35} - \frac{18}{5} = \frac{51}{35} - \frac{18 \cdot 7}{35} = \frac{51 - 126}{35} = -\frac{75}{35} = -\frac{15}{7} \)
2. б) \( \frac{1}{13} \cdot \left( 2 \frac{3}{8} - 1 \frac{5}{6} \right) \cdot 2 \frac{2}{5} + \frac{9}{10} = \frac{1}{13} \cdot \left( \frac{19}{8} - \frac{11}{6} \right) \cdot \frac{12}{5} + \frac{9}{10} = \frac{1}{13} \cdot \left( \frac{19 \cdot 3}{24} - \frac{11 \cdot 4}{24} \right) \cdot \frac{12}{5} + \frac{9}{10} = \frac{1}{13} \cdot \left( \frac{57 - 44}{24} \right) \cdot \frac{12}{5} + \frac{9}{10} = \frac{1}{13} \cdot \frac{13}{24} \cdot \frac{12}{5} + \frac{9}{10} = \frac{1 \cdot 1 \cdot 12 \cdot 1}{1 \cdot 24 \cdot 5 \cdot 1} + \frac{9}{10} = \frac{12}{120} + \frac{9}{10} = \frac{1}{10} + \frac{9}{10} = \frac{10}{10} = 1 \)
3. в) \( 4 \frac{3}{4} - \left( 1 \frac{2}{5} \cdot 2 \frac{1}{2} \right) \cdot 1 \frac{3}{5} = \frac{19}{4} - \left( \frac{7}{5} \cdot \frac{5}{2} \right) \cdot \frac{8}{5} = \frac{19}{4} - \frac{7}{2} \cdot \frac{8}{5} = \frac{19}{4} - \frac{7 \cdot 4}{5} = \frac{19}{4} - \frac{28}{5} = \frac{19 \cdot 5}{20} - \frac{28 \cdot 4}{20} = \frac{95 - 112}{20} = -\frac{17}{20} \)
4. г) \( 1 \frac{1}{2} \cdot \frac{2}{9} - \left( 6 - 5 \frac{3}{5} \right)^2 = \frac{3}{2} \cdot \frac{2}{9} - \left( 6 - \frac{28}{5} \right)^2 = \frac{3 \cdot 1}{1 \cdot 9} - \left( \frac{30}{5} - \frac{28}{5} \right)^2 = \frac{1}{3} - \left( \frac{2}{5} \right)^2 = \frac{1}{3} - \frac{4}{25} = \frac{1 \cdot 25}{75} - \frac{4 \cdot 3}{75} = \frac{25 - 12}{75} = \frac{13}{75} \)
5. д) \( 4 \frac{1}{15} - 3 \frac{9}{10} \right) \cdot 6 \frac{6}{7} + 2 = \left( \frac{61}{15} - \frac{39}{10} \right) \cdot \frac{48}{7} + 2 = \left( \frac{61 \cdot 2}{30} - \frac{39 \cdot 3}{30} \right) \cdot \frac{48}{7} + 2 = \left( \frac{122 - 117}{30} \right) \cdot \frac{48}{7} + 2 = \frac{5}{30} \cdot \frac{48}{7} + 2 = \frac{1}{6} \cdot \frac{48}{7} + 2 = \frac{8}{7} + 2 = \frac{8}{7} + \frac{14}{7} = \frac{22}{7} \)
6. е) \( \frac{1}{3} \cdot \left( 4 - \left( 1 \frac{1}{2} \right)^3 \right) \cdot \left( 3 - 2 \frac{7}{15} \right) = \frac{1}{3} \cdot \left( 4 - \left( \frac{3}{2} \right)^3 \right) \cdot \left( 3 - \frac{37}{15} \right) = \frac{1}{3} \cdot \left( 4 - \frac{27}{8} \right) \cdot \left( \frac{45}{15} - \frac{37}{15} \right) = \frac{1}{3} \cdot \left( \frac{32}{8} - \frac{27}{8} \right) \cdot \frac{8}{15} = \frac{1}{3} \cdot \frac{5}{8} \cdot \frac{8}{15} = \frac{1 \cdot 5 \cdot 8}{3 \cdot 8 \cdot 15} = \frac{5}{45} = \frac{1}{9} \)

Ответ: а) -\(\frac{15}{7}\); б) 1; в) -\(\frac{17}{20}\); г) \(\frac{13}{75}\); д) \(\frac{22}{7}\); е) \(\frac{1}{9}\).