2. Вычислите: а) \(\frac{55}{48} : (\frac{11}{16} + \frac{3}{32}) - \frac{14}{15} \cdot \frac{5}{7}\); б) \((\frac{1}{12} + \frac{1}{13})^2 : (\frac{1}{12} - \frac{1}{13})^2 \cdot (\frac{1}{10})^3\).

Решение:

1. а) \(\frac{55}{48} : (\frac{11}{16} + \frac{3}{32}) - \frac{14}{15} \cdot \frac{5}{7} =\)

1) \(\frac{11}{16} + \frac{3}{32} = \frac{11 \cdot 2}{16 \cdot 2} + \frac{3}{32} = \frac{22}{32} + \frac{3}{32} = \frac{22 + 3}{32} = \frac{25}{32}\)

2) \(\frac{55}{48} : \frac{25}{32} = \frac{55}{48} \cdot \frac{32}{25} = \frac{55 \cdot 32}{48 \cdot 25} = \frac{11 \cdot 5 \cdot 16 \cdot 2}{16 \cdot 3 \cdot 5 \cdot 5} = \frac{11 \cdot 2}{3 \cdot 5} = \frac{22}{15}\)

3) \(\frac{14}{15} \cdot \frac{5}{7} = \frac{14 \cdot 5}{15 \cdot 7} = \frac{2 \cdot 7 \cdot 5}{3 \cdot 5 \cdot 7} = \frac{2}{3}\)

4) \(\frac{22}{15} - \frac{2}{3} = \frac{22}{15} - \frac{2 \cdot 5}{3 \cdot 5} = \frac{22}{15} - \frac{10}{15} = \frac{22 - 10}{15} = \frac{12}{15} = \frac{4 \cdot 3}{5 \cdot 3} = \frac{4}{5}\)

Ответ: \(\frac{4}{5}\)

1. б) \((\frac{1}{12} + \frac{1}{13})^2 : (\frac{1}{12} - \frac{1}{13})^2 \cdot (\frac{1}{10})^3 =\)

1) \((\frac{1}{12} + \frac{1}{13})^2 = (\frac{13}{12 \cdot 13} + \frac{12}{13 \cdot 12})^2 = (\frac{13 + 12}{156})^2 = (\frac{25}{156})^2 = \frac{25^2}{156^2} = \frac{625}{24336}\)

2) \((\frac{1}{12} - \frac{1}{13})^2 = (\frac{13}{12 \cdot 13} - \frac{12}{13 \cdot 12})^2 = (\frac{13 - 12}{156})^2 = (\frac{1}{156})^2 = \frac{1}{156^2} = \frac{1}{24336}\)

3) \(( \frac{25}{156})^2 : (\frac{1}{156})^2 = \frac{25^2}{156^2} : \frac{1}{156^2} = \frac{25^2}{156^2} \cdot \frac{156^2}{1} = 25^2 = 625\)

4) \((\frac{1}{10})^3 = \frac{1^3}{10^3} = \frac{1}{1000}\)

5) \(625 \cdot \frac{1}{1000} = \frac{625}{1000} = \frac{5 \cdot 125}{5 \cdot 200} = \frac{5 \cdot 5 \cdot 25}{5 \cdot 5 \cdot 40} = \frac{25}{40} = \frac{5 \cdot 5}{5 \cdot 8} = \frac{5}{8}\)

Ответ: \(\frac{5}{8}\)