5. Решите уравнение (8⋅x³)⁵(3x³)⁴ (9x⁶)⁴ = 24.

$$\frac{(8x^3)^5(3x^3)^4}{(9x^6)^4} = 24$$
$$\frac{8^5 \cdot (x^3)^5 \cdot 3^4 \cdot (x^3)^4}{9^4 \cdot (x^6)^4} = 24$$
$$\frac{8^5 \cdot x^{15} \cdot 3^4 \cdot x^{12}}{9^4 \cdot x^{24}} = 24$$
$$\frac{(2^3)^5 \cdot 3^4 \cdot x^{15+12}}{(3^2)^4 \cdot x^{24}} = 24$$
$$\frac{2^{15} \cdot 3^4 \cdot x^{27}}{3^8 \cdot x^{24}} = 24$$
$$2^{15} \cdot 3^{4-8} \cdot x^{27-24} = 24$$
$$2^{15} \cdot 3^{-4} \cdot x^3 = 24$$
$$\frac{2^{15}}{3^4} \cdot x^3 = 24$$
$$x^3 = \frac{24 \cdot 3^4}{2^{15}} = \frac{3 \cdot 8 \cdot 3^4}{2^{15}} = \frac{3^5 \cdot 2^3}{2^{15}} = \frac{3^5}{2^{12}}$$
$$x = \sqrt[3]{\frac{3^5}{2^{12}}} = \frac{\sqrt[3]{3^5}}{\sqrt[3]{2^{12}}} = \frac{3 \cdot \sqrt[3]{3^2}}{2^4} = \frac{3 \cdot \sqrt[3]{9}}{16}$$

Ответ: $$x = \frac{3 \cdot \sqrt[3]{9}}{16}$$