11. Given positive numbers $a, b, c, d, e, f$, which simultaneously satisfy
$$
\begin{array}{l}
\frac{b c d e f}{a}=\frac{1}{2}, \frac{a c d e f}{b}=\frac{1}{4}, \frac{a b d e f}{c}=\frac{1}{8}, \\
\frac{a b c e f}{d}=2, \frac{a b c d f}{e}=4, \frac{a b c d e}{f}=8 .
\end{array}
$$

Then the value of $a+b+c+d+e+f$ is $\qquad$
11. $\frac{5}{2}+\frac{15 \sqrt{2}}{4}$.
Multiplying both sides of the 6 equations yields $(a b c d e f)^{4}=1$. Thus, $a b c d e f=1$, $a^{2}=2, b^{2}=4, c^{2}=8, d^{2}=\frac{1}{2}, e^{2}=\frac{1}{4}, f^{2}=\frac{1}{8}$.
Therefore, $a+b+c+d+e+f$
$$
\begin{array}{l}
=\sqrt{2}+\sqrt{4}+\sqrt{8}+\sqrt{\frac{1}{2}}+\sqrt{\frac{1}{4}}+\sqrt{\frac{1}{8}} \\
=\frac{5}{2}+\frac{15 \sqrt{2}}{4} .
\end{array}
$$