7. If $z$ is a complex number with a non-zero real part, then the minimum value of $\frac{\operatorname{Re}\left(z^{4}\right)}{(\operatorname{Re}(z))^{4}}$ is $\qquad$ . (Wang Yongxi)

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7. -8 .

Let $z=a+b \mathrm{i}(a, b \in \mathbf{R}, a \neq 0)$, then it is easy to know that $\operatorname{Re}\left(z^{4}\right)=a^{4}-6 a^{2} b^{2}+b^{4}$. Therefore,
$$
\frac{\operatorname{Re}\left(z^{4}\right)}{(\operatorname{Re}(z))^{4}}=\frac{a^{4}-6 a^{2} b^{2}+b^{4}}{a^{4}}=t^{2}-6 t-1=(t-3)^{2}-8 \geqslant 8,
$$

Here, $t=\frac{b^{2}}{a^{2}} \geqslant 0$. The equality holds if and only if $t=3$, i.e., $b^{2}=3 a^{2}$. Therefore, the minimum value sought is -8.