7. Найдите значение выражения x⁵y - xy⁵ / 5(3y - x) ⋅ 2(x - 3y) / x⁴ - y⁴ при x = -1/7 и y = -14.

Упростим выражение: (xy(x⁴ - y⁴)) / (5(3y - x)) ⋅ 2(x - 3y) / (x² - y²)(x² + y²)

= (xy(x² - y²)(x² + y²)) / (5(3y - x)) ⋅ 2(x - 3y) / ((x - y)(x + y)(x² + y²))

= (xy) / (5(3y - x)) ⋅ 2(x - 3y) / (x - y)(x + y)

Подставим значения x = -1/7 и y = -14:

3y - x = 3(-14) - (-1/7) = -42 + 1/7 = -293/7

x - 3y = -1/7 - 3(-14) = -1/7 + 42 = 293/7

xy = (-1/7)(-14) = 2

x - y = -1/7 - (-14) = -1/7 + 14 = 97/7

x + y = -1/7 + (-14) = -99/7

Подставляем в упрощенное выражение: (2) / (5(-293/7)) ⋅ 2(293/7) / ((97/7)(-99/7))

= (2) / (-1465/7) ⋅ (586/7) / (-9603/49)

= (14/ -1465) ⋅ (586/7) ⋅ (-49/9603)

= (-14/1465) ⋅ (586/7) ⋅ (-49/9603)

= (-2) ⋅ (586/1465) ⋅ (-49/9603)

= (-1172/1465) ⋅ (-49/9603)

Перегруппируем: (xy * 2 * (x-3y)) / (5(3y-x) * (x-y)(x+y)) = (2xy(x-3y)) / (5(3y-x)(x-y)(x+y))

Заметим, что x - 3y = -(3y - x). Подставляем:

(2xy * -(3y-x)) / (5(3y-x)(x-y)(x+y)) = -2xy / (5(x-y)(x+y)) = -2xy / (5(x² - y²))

x = -1/7, y = -14

xy = (-1/7)(-14) = 2

x² = 1/49

y² = 196

x² - y² = 1/49 - 196 = (1 - 196*49)/49 = (1 - 9604)/49 = -9603/49

-2xy / (5(x² - y²)) = -2(2) / (5(-9603/49)) = -4 / (-48015/49) = 4 * 49 / 48015 = 196 / 48015

Проверка упрощения: x⁵y - xy⁵ = xy(x⁴ - y⁴) = xy(x² - y²)(x² + y²)

5(3y - x) ⋅ (x⁴ - y⁴) / (x⁴ - y⁴) = 5(3y - x)

2(x - 3y) / (x⁴ - y⁴)

Выражение: (xy(x⁴ - y⁴)) / (5(3y - x)) ⋅ 2(x - 3y) / (x⁴ - y⁴)

= (xy * 2(x - 3y)) / (5(3y - x)) = (2xy(x - 3y)) / (5(3y - x))

Так как x - 3y = -(3y - x), то выражение равно (2xy * -(3y - x)) / (5(3y - x)) = -2xy / 5.

Подставляем x = -1/7 и y = -14:

-2 * (-1/7) * (-14) / 5 = -2 * (2) / 5 = -4/5.