5. Найдите значение выражения \frac{x³ + x²y + xy²}{10(y - 2x)} \cdot \frac{3(2x - y)}{x + y} при x = -\frac{1}{9} и y = -9.

5. Найдем значение выражения \frac{x³ + x²y + xy²}{10(y - 2x)} \cdot \frac{3(2x - y)}{x + y} при x = -\frac{1}{9} и y = -9.

Преобразуем выражение:

$$\frac{x^3 + x^2y + xy^2}{10(y - 2x)} \cdot \frac{3(2x - y)}{x + y} = \frac{x(x^2 + xy + y^2)}{10(y - 2x)} \cdot \frac{3(2x - y)}{x + y} $$

Подставим значения x = -\frac{1}{9} и y = -9:

$$\frac{-\frac{1}{9}((-\frac{1}{9})^2 + (-\frac{1}{9})(-9) + (-9)^2)}{10(-9 - 2(-\frac{1}{9}))} \cdot \frac{3(2(-\frac{1}{9}) - (-9))}{-\frac{1}{9} - 9} = \frac{-\frac{1}{9}(\frac{1}{81} + 1 + 81)}{10(-\frac{81}{9} + \frac{2}{9})} \cdot \frac{3(-\frac{2}{9} + 9)}{-\frac{1}{9} - \frac{81}{9}} = \frac{-\frac{1}{9}(\frac{1 + 81 + 6561}{81})}{10(-\frac{79}{9})} \cdot \frac{3(\frac{-2 + 81}{9})}{-\frac{82}{9}} = \frac{-\frac{1}{9}(\frac{6643}{81})}{-\frac{790}{9}} \cdot \frac{3(\frac{79}{9})}{-\frac{82}{9}} = \frac{-\frac{6643}{729}}{-\frac{790}{9}} \cdot \frac{\frac{237}{9}}{-\frac{82}{9}} = \frac{6643}{729} \cdot \frac{9}{790} \cdot \frac{237}{9} \cdot \frac{9}{82} = \frac{6643}{81} \cdot \frac{1}{790} \cdot \frac{237}{1} \cdot \frac{1}{82} = \frac{6643 \cdot 237}{81 \cdot 790 \cdot 82} = \frac{1574491}{5317380} \approx 0.296 $$

При x = -\frac{1}{9} и y = -9, x+y = -\frac{1}{9} - 9 = -\frac{82}{9}.

Рассмотрим выражение x³ + x²y + xy² = x(x² + xy + y²). При x = -\frac{1}{9} и y = -9, x² + xy + y² = \frac{1}{81} + 1 + 81 = \frac{1+81+6561}{81} = \frac{6643}{81} = 82\frac{1}{81}.

Выражение \frac{3(2x - y)}{x+y} = \frac{3(-\frac{2}{9} + 9)}{-\frac{1}{9} - 9} = \frac{3(\frac{79}{9})}{-\frac{82}{9}} = \frac{\frac{79}{3}}{-\frac{82}{9}} = \frac{79}{3} \cdot \frac{-9}{82} = \frac{79}{1} \cdot \frac{-3}{82} = \frac{-237}{82}

Выражение 10(y - 2x) = 10(-9 + \frac{2}{9}) = 10(\frac{-81+2}{9}) = 10(\frac{-79}{9}) = \frac{-790}{9}.

Итоговое выражение \frac{x(x^2 + xy + y^2)}{10(y - 2x)} \cdot \frac{3(2x - y)}{x + y} = \frac{-\frac{1}{9} \cdot \frac{6643}{81}}{\frac{-790}{9}} \cdot \frac{\frac{-237}{82}}{1} = \frac{-\frac{6643}{729}}{\frac{-790}{9}} \cdot \frac{-237}{82} = \frac{6643}{729} \cdot \frac{9}{790} \cdot \frac{-237}{82} = \frac{6643}{81} \cdot \frac{1}{790} \cdot \frac{-237}{82} = \frac{-1574391}{5317380} \approx -0.296

Ответ: \frac{-1574391}{5317380} \approx -0.296