П.17 Найдите значение выражения: а) \( \frac{12}{17} : \left(-1\frac{7}{17}\right) + 5,88 : (-14,7) - 0,1 \); б) \( \left(8-5\frac{5}{8}\right) \cdot 2\frac{2}{3} + \left(8-6\frac{3}{4}\right) : 1\frac{3}{4} \); в) \( 5,5 - 3\frac{1}{3} \cdot \left(1\frac{2}{3} + 1\frac{2}{5}\right) : 2\frac{4}{5} \); г) \( 2\frac{4}{5} : 1\frac{3}{5} \cdot 2\frac{1}{2} - 4\frac{2}{7} \cdot \frac{7}{15} \cdot \left(1\frac{1}{3}\right)^3 \); д) \( \frac{5}{8} \cdot \frac{4}{15} - \frac{14}{33} : \left(\frac{7}{11} + \frac{1}{12}\right) \); е) \( \frac{2}{7} \cdot \left(3\frac{1}{2}\right)^2 - \frac{5}{13} : 3\frac{1}{13} + \frac{9}{10} : 3\frac{3}{5} \).

Решение:

а) \( \frac{12}{17} : \left(-1\frac{7}{17}\right) + 5,88 : (-14,7) - 0,1 \)

1. \( -1\frac{7}{17} = -\frac{17+7}{17} = -\frac{24}{17} \)
2. \( \frac{12}{17} : \left(-\frac{24}{17}\right) = \frac{12}{17} \cdot \left(-\frac{17}{24}\right) = -\frac{12}{24} = -\frac{1}{2} = -0,5 \)
3. \( 5,88 : (-14,7) = -0,4 \)
4. \( -0,5 + (-0,4) - 0,1 = -0,5 - 0,4 - 0,1 = -1 \)

б) \( \left(8-5\frac{5}{8}\right) \cdot 2\frac{2}{3} + \left(8-6\frac{3}{4}\right) : 1\frac{3}{4} \)

1. \( 8 - 5\frac{5}{8} = 8 - \frac{45}{8} = \frac{64-45}{8} = \frac{19}{8} \)
2. \( 2\frac{2}{3} = \frac{8}{3} \)
3. \( \frac{19}{8} \cdot \frac{8}{3} = \frac{19}{3} \)
4. \( 8 - 6\frac{3}{4} = 8 - \frac{27}{4} = \frac{32-27}{4} = \frac{5}{4} \)
5. \( 1\frac{3}{4} = \frac{7}{4} \)
6. \( \frac{5}{4} : \frac{7}{4} = \frac{5}{4} \cdot \frac{4}{7} = \frac{5}{7} \)
7. \( \frac{19}{3} + \frac{5}{7} = \frac{19 \cdot 7 + 5 \cdot 3}{21} = \frac{133+15}{21} = \frac{148}{21} \)

в) \( 5,5 - 3\frac{1}{3} \cdot \left(1\frac{2}{3} + 1\frac{2}{5}\right) : 2\frac{4}{5} \)

1. \( 1\frac{2}{3} = \frac{5}{3} \)
2. \( 1\frac{2}{5} = \frac{7}{5} \)
3. \( \frac{5}{3} + \frac{7}{5} = \frac{25+21}{15} = \frac{46}{15} \)
4. \( 3\frac{1}{3} = \frac{10}{3} \)
5. \( \frac{10}{3} \cdot \frac{46}{15} = \frac{10 \cdot 46}{3 \cdot 15} = \frac{2 \cdot 46}{3 \cdot 3} = \frac{92}{9} \)
6. \( 2\frac{4}{5} = \frac{14}{5} \)
7. \( \frac{92}{9} : \frac{14}{5} = \frac{92}{9} \cdot \frac{5}{14} = \frac{46 \cdot 5}{9 \cdot 7} = \frac{230}{63} \)
8. \( 5,5 = \frac{11}{2} \)
9. \( \frac{11}{2} - \frac{230}{63} = \frac{11 \cdot 63 - 230 \cdot 2}{126} = \frac{693 - 460}{126} = \frac{233}{126} \)

г) \( 2\frac{4}{5} : 1\frac{3}{5} \cdot 2\frac{1}{2} - 4\frac{2}{7} \cdot \frac{7}{15} \cdot \left(1\frac{1}{3}\right)^3 \)

1. \( 2\frac{4}{5} = \frac{14}{5} \)
2. \( 1\frac{3}{5} = \frac{8}{5} \)
3. \( \frac{14}{5} : \frac{8}{5} = \frac{14}{5} \cdot \frac{5}{8} = \frac{14}{8} = \frac{7}{4} \)
4. \( 2\frac{1}{2} = \frac{5}{2} \)
5. \( \frac{7}{4} \cdot \frac{5}{2} = \frac{35}{8} \)
6. \( 4\frac{2}{7} = \frac{30}{7} \)
7. \( \frac{30}{7} \cdot \frac{7}{15} = \frac{30}{15} = 2 \)
8. \( 1\frac{1}{3} = \frac{4}{3} \)
9. \( \left(\frac{4}{3}\right)^3 = \frac{64}{27} \)
10. \( 2 \cdot \frac{64}{27} = \frac{128}{27} \)
11. \( \frac{35}{8} - \frac{128}{27} = \frac{35 \cdot 27 - 128 \cdot 8}{216} = \frac{945 - 1024}{216} = -\frac{79}{216} \)

д) \( \frac{5}{8} \cdot \frac{4}{15} - \frac{14}{33} : \left(\frac{7}{11} + \frac{1}{12}\right) \)

1. \( \frac{5}{8} \cdot \frac{4}{15} = \frac{5 \cdot 4}{8 \cdot 15} = \frac{20}{120} = \frac{1}{6} \)
2. \( \frac{7}{11} + \frac{1}{12} = \frac{7 \cdot 12 + 1 \cdot 11}{132} = \frac{84+11}{132} = \frac{95}{132} \)
3. \( \frac{14}{33} : \frac{95}{132} = \frac{14}{33} \cdot \frac{132}{95} = \frac{14 \cdot 4}{1 \cdot 95} = \frac{56}{95} \)
4. \( \frac{1}{6} - \frac{56}{95} = \frac{1 \cdot 95 - 56 \cdot 6}{570} = \frac{95 - 336}{570} = -\frac{241}{570} \)

е) \( \frac{2}{7} \cdot \left(3\frac{1}{2}\right)^2 - \frac{5}{13} : 3\frac{1}{13} + \frac{9}{10} : 3\frac{3}{5} \)

1. \( 3\frac{1}{2} = \frac{7}{2} \)
2. \( \left(\frac{7}{2}\right)^2 = \frac{49}{4} \)
3. \( \frac{2}{7} \cdot \frac{49}{4} = \frac{2 \cdot 49}{7 \cdot 4} = \frac{1 \cdot 7}{1 \cdot 2} = \frac{7}{2} \)
4. \( 3\frac{1}{13} = \frac{40}{13} \)
5. \( \frac{5}{13} : \frac{40}{13} = \frac{5}{13} \cdot \frac{13}{40} = \frac{5}{40} = \frac{1}{8} \)
6. \( 3\frac{3}{5} = \frac{18}{5} \)
7. \( \frac{9}{10} : \frac{18}{5} = \frac{9}{10} \cdot \frac{5}{18} = \frac{1 \cdot 1}{2 \cdot 2} = \frac{1}{4} \)
8. \( \frac{7}{2} - \frac{1}{8} + \frac{1}{4} = \frac{28 - 1 + 2}{8} = \frac{29}{8} \)

Ответ: а) -1; б) \(\frac{148}{21}\); в) \(\frac{233}{126}\); г) \(-\frac{79}{216}\); д) \(-\frac{241}{570}\); е) \(\frac{29}{8}\).