7.060. $\sqrt{2^{x} \cdot \sqrt[3]{4^{x} \cdot 0.125^{1 / x}}}=4 \sqrt[3]{2}$.
Solution.

Domain of definition: $x \neq 0$.

Rewrite the equation as

$$
2^{\frac{x}{2}} \cdot 2^{\frac{x}{3}} \cdot 2^{-\frac{1}{2 x}}=2^{2} \cdot 2^{\frac{1}{3}}, 2^{\frac{x}{2}+\frac{x}{3}-\frac{1}{2 x}}=2^{2+\frac{1}{3}}
$$

from which

$\frac{x}{2}+\frac{x}{3}-\frac{1}{2 x}=\frac{7}{3}, 5 x^{2}-14 x-3=0$.

Then $x_{1}=-\frac{1}{5}, x_{2}=3$.

Answer: $-\frac{1}{5} ; 3$.