10) Найдите значение выражения (5х2 + 2y³) (2y3 - 5x2) при х² = 1, y² = 2.

Из условия x⁴ = 1/5, значит x² = ± \(\sqrt{1/5}\). Так как y² = 2, то y = ± \(\sqrt{2}\). Тогда y³ = y² ⋅ y = ±2\(\sqrt{2}\).

Подставим значения в выражение: (5x² + 2y³) (2y³ - 5x²)

Рассмотрим случай x² = \(\sqrt{1/5}\) и y³ = 2\(\sqrt{2}\):

(5(\(\sqrt{1/5}\)) + 2(2\(\sqrt{2}\))) (2(2\(\sqrt{2}\)) - 5(\(\sqrt{1/5}\))) = (\(\sqrt{5}\) + 4\(\sqrt{2}\)) (4\(\sqrt{2}\) - \(\sqrt{5}\)) = (4\(\sqrt{2}\))² - (\(\sqrt{5}\))² = 16 ⋅ 2 - 5 = 32 - 5 = 27

Рассмотрим случай x² = -\(\sqrt{1/5}\) и y³ = -2\(\sqrt{2}\):

(5(-\(\sqrt{1/5}\)) + 2(-2\(\sqrt{2}\))) (2(-2\(\sqrt{2}\)) - 5(-\(\sqrt{1/5}\))) = (-\(\sqrt{5}\) - 4\(\sqrt{2}\)) (-4\(\sqrt{2}\) + \(\sqrt{5}\)) = (\(\sqrt{5}\) + 4\(\sqrt{2}\)) (\(\sqrt{5}\) - 4\(\sqrt{2}\)) = (\(\sqrt{5}\))² - (4\(\sqrt{2}\))² = 5 - 16 ⋅ 2 = 5 - 32 = -27

Ответ: ±27