7. Let the 10 complex roots of the equation $x^{10}+(13 x-1)^{10}=0$ be $x_{1}, x_{2}, \cdots, x_{10}$. Then
$$
\frac{1}{x_{1} \overline{x_{1}}}+\frac{1}{x_{2} \overline{x_{2}}}+\cdots+\frac{1}{x_{5} \overline{x_{5}}}=
$$
$\qquad$
7. 850 .

Let $\varepsilon=\cos \frac{\pi}{10}+\mathrm{i} \sin \frac{\pi}{10}$. Then $\varepsilon^{10}=-1$.
Given that the 10 complex roots of the equation $x^{10}+(13 x-1)^{10}=0$ are $x_{1}, x_{2}, \cdots, x_{10}$, we can assume them to be $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$, $\overline{x_{1}}, \overline{x_{2}}, \overline{x_{3}}, \overline{x_{4}}, \overline{x_{5}}$.
From $(13 x-1)^{10}=-x^{10}$, we know
$$
13 x_{k}-1=x_{k} \varepsilon^{2 k-1}(k=1,2, \cdots, 5) \text {. }
$$

Thus, $\frac{1}{x_{k}}=13-\varepsilon^{2 k-1}$.
$$
\begin{array}{l}
\text { Hence } \frac{1}{x_{1} \overline{x_{1}}}+\frac{1}{x_{2} \overline{x_{2}}}+\cdots+\frac{1}{x_{5} \overline{x_{5}}} \\
=\sum_{k=1}^{5}\left(13-\varepsilon^{2 k-1}\right)\left(13-\bar{\varepsilon}^{2 k-1}\right) \\
=\sum_{k=1}^{5}\left[170-13\left(\varepsilon^{2 k-1}+\bar{\varepsilon}^{2 k-1}\right)\right] \\
=850-13 \sum_{k=1}^{5}\left(\varepsilon^{2 k-1}+\bar{\varepsilon}^{2 k-1}\right)=850 .
\end{array}
$$