7.243. $5^{x} \cdot \sqrt[x]{8^{x-1}}=500$.
Solution.
Rewrite the equation as $5^{x} \cdot 8^{\frac{x-1}{x}}=500 \Leftrightarrow \frac{5^{x} \cdot 8}{8^{1 / x}}=500 \Leftrightarrow$ $\Leftrightarrow \frac{5^{x} \cdot 2}{8^{1 / x}}=125 \Leftrightarrow 5^{x-3}=2^{3 / x-1} \Rightarrow\left\{\begin{array}{l}x-3=0, \\ \frac{3}{x}-1=0,\end{array}\right.$ from which $x=3$.

Answer: 3.