3. Let $z_{1}, z_{2}, \cdots, z_{12}$ be the roots of the equation $z^{12}=2^{36}$. Let $w_{j} (j=1,2, \cdots, 12)$ be one of $z_{j}$ and i $z_{j}$. Then the maximum value of $\operatorname{Re} \sum_{j=1}^{12} w_{j}$ is . $\qquad$
3. $16(1+\sqrt{3})$.

As shown in Figure 2, using the geometric meaning, $y=x$, $y=-x$ and the coordinate axes divide the circle $x^{2}+y^{2}=8$ into eight parts.

When $z_{j}$ is above the line $y=-x$, $w_{j}=z_{j}$; otherwise, $w_{j}=\mathrm{i} z_{j}$.
$$
\begin{array}{l}
\text { Hence Re } \sum_{j=1}^{12} w_{j} \\
=2^{3} \sum_{j=-1}^{4} \cos \frac{2 j \pi}{12}-2^{3} \sum_{j=5}^{10} \cos \frac{2 j \pi}{12} \\
=16(1+\sqrt{3}) .
\end{array}
$$