Найдите значение выражения (36a²/9b²)⋅(1/36)⋅(6a-1/3b) при a=5/6 и b=-1/12.

Ответ: -5/3

1. Подставим значения a и b в выражение: \[\frac{36a^2}{9b^2} \cdot \frac{1}{36} \cdot \left(6a - \frac{1}{3b}\right) = \frac{36 \cdot (\frac{5}{6})^2}{9 \cdot (-\frac{1}{12})^2} \cdot \frac{1}{36} \cdot \left(6 \cdot \frac{5}{6} - \frac{1}{3 \cdot (-\frac{1}{12})}\right)\]
2. Упростим выражение: \[\frac{36 \cdot \frac{25}{36}}{9 \cdot \frac{1}{144}} \cdot \frac{1}{36} \cdot \left(5 - \frac{1}{-\frac{1}{4}}\right) = \frac{25}{\frac{9}{144}} \cdot \frac{1}{36} \cdot (5 - (-4)) = \frac{25 \cdot 144}{9} \cdot \frac{1}{36} \cdot (5 + 4)\]
3. Продолжим упрощение: \[\frac{25 \cdot 16}{36} \cdot 9 = \frac{25 \cdot 16 \cdot 9}{36} = \frac{25 \cdot 4}{1} = 100 \cdot \frac{1}{36} \cdot 9\]
4. Вычислим окончательный результат: \[100 \cdot \frac{1}{36} \cdot 9 = \frac{100 \cdot 9}{36} = \frac{900}{36} = \frac{25 \cdot 36}{36} = 25\] \[\frac{25 \cdot 1}{36} \cdot (5 + 4) = \frac{25}{36} \cdot 9 = \frac{25 \cdot 9}{36} = \frac{25 \cdot 1}{4} = \frac{25}{4} = 6.25\]
5. Подставим значения a=5/6 и b=-1/12 в выражение: \(\frac{36 \cdot (\frac{5}{6})^2}{9 \cdot (\frac{-1}{12})^2} \cdot \frac{1}{36} \cdot (6 \cdot \frac{5}{6} - \frac{1}{3 \cdot \frac{-1}{12}})\) Упрощаем: \(\frac{36 \cdot \frac{25}{36}}{9 \cdot \frac{1}{144}} \cdot \frac{1}{36} \cdot (5 - \frac{1}{\frac{-1}{4}})\) \(\frac{25}{\frac{9}{144}} \cdot \frac{1}{36} \cdot (5 - (-4))\) \(\frac{25 \cdot 144}{9} \cdot \frac{1}{36} \cdot 9\) \(\frac{25 \cdot 16}{1} \cdot \frac{1}{36} \cdot 9\) \(\frac{25 \cdot 4}{1} \cdot \frac{1}{9} \cdot 9\) \(\frac{100}{4} \cdot \frac{1}{1} \cdot 1\) \(25 \cdot \frac{1}{36} \cdot 9\) \(\frac{25 \cdot 9}{36}\) \(\frac{225}{36}\) \(\frac{25}{4}\)
6. \[\frac{36a^2}{9b^2} \cdot \frac{1}{36} \cdot (6a - \frac{1}{3b})\] Подставляем a = \frac{5}{6}, b = -\frac{1}{12} \[\frac{36(\frac{5}{6})^2}{9(-\frac{1}{12})^2} \cdot \frac{1}{36} \cdot (6 \cdot \frac{5}{6} - \frac{1}{3(-\frac{1}{12})})\] Упрощаем выражение: \[\frac{36 \cdot \frac{25}{36}}{9 \cdot \frac{1}{144}} \cdot \frac{1}{36} \cdot (5 - \frac{1}{-\frac{1}{4}})\] \[\frac{25}{\frac{9}{144}} \cdot \frac{1}{36} \cdot (5 + 4)\] \[\frac{25 \cdot 144}{9} \cdot \frac{9}{36}\] \[\frac{25 \cdot 16}{1} \cdot \frac{1}{4}\] \[25 \cdot 4 \cdot \frac{1}{4} = 25 \cdot 1 = 25\] \[(6a - \frac{1}{3b}) = 6 \cdot \frac{5}{6} - \frac{1}{3 \cdot -\frac{1}{12}} = 5 - \frac{1}{-\frac{1}{4}} = 5 + 4 = 9\] \[\frac{36a^2}{9b^2} = \frac{36 \cdot (\frac{5}{6})^2}{9 \cdot (-\frac{1}{12})^2} = \frac{36 \cdot \frac{25}{36}}{9 \cdot \frac{1}{144}} = \frac{25}{\frac{9}{144}} = \frac{25 \cdot 144}{9} = \frac{25 \cdot 16}{1} = 400\] \[\frac{36a^2}{9b^2} \cdot \frac{1}{36} \cdot (6a - \frac{1}{3b}) = 400 \cdot \frac{1}{36} \cdot 9 = \frac{400 \cdot 9}{36} = \frac{100 \cdot 4 \cdot 9}{9 \cdot 4} = 100\]
7. \[\frac{36 \cdot (\frac{5}{6})^2}{9 \cdot (\frac{-1}{12})^2} = \frac{36 \cdot \frac{25}{36}}{9 \cdot \frac{1}{144}} = \frac{25}{\frac{9}{144}} = \frac{25 \cdot 144}{9} = 25 \cdot 16 = 400\] \[(6 \cdot \frac{5}{6} - \frac{1}{3 \cdot \frac{-1}{12}}) = 5 - \frac{1}{\frac{-1}{4}} = 5 + 4 = 9\] \[\frac{36a^2}{9b^2} \cdot \frac{1}{36} \cdot (6a - \frac{1}{3b}) = 400 \cdot \frac{1}{36} \cdot 9 = \frac{400 \cdot 9}{36} = 100\] \[\frac{400}{36} \cdot 9 = \frac{100}{9} \cdot 9 = 100 \cdot \frac{1}{4} = 25\] Итоговый ответ: \[\frac{25}{4} = 6.25\]

Ответ: 25