X^ (-4/3) - x^ (-2/3) - 8=0

1. Обозначим:

x^ (-2/3) = z; x^ (-4/3) - x^ (-2/3) - 8 = 0; (x^ (-2/3)) ^2 - x^ (-2/3) - 8 = 0.

2. Квадратное уравнение:

z^2 - z - 8 = 0; D = 1^2 + 4 * 8 = 33; z = (1 ± √33) / 2;

1) z = (1 - √33) / 2;

x^ (-2/3) = (1 - √33) / 2 < 0, нет решения;

2) z = (1 + √33) / 2;

x^ (-2/3) = (1 + √33) / 2; 1/x^ (2/3) = (1 + √33) / 2; x^ (2/3) = 2 / (√33 + 1) = 2 (√33 - 1) / (√33 + 1) (√33 - 1) = 2 (√33 - 1) / (33 - 1) = (√33 - 1) / 16; x^2 = (√33 - 1) ^3/2^12 = (33√33 - 3 * 33 + 3√33 - 1) / 2^12 = (36√33 - 100) / 2^12 = (9√33 - 25) / 2^10; x = ±√ (9√33 - 25) / 32.

Ответ: ±√ (9√33 - 25) / 32.