If $\bar{a}, \bar{b}, \bar{c}$ are three coplanar vectors such that $|\bar{a}|=1,|\bar{b}|=2$, $\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}=8$, the angle between $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ is $45^{\circ}$, then $|\bar{a} \times(\bar{b} \times \bar{c})|=$

Given vectors $\bar{a}$, $\bar{b}$, $\bar{c}$ are coplanar with $|\bar{a}| = 1$, $|\bar{b}| = 2$, $\bar{b} \cdot \bar{c} = 8$, and the angle between $\bar{b}$ and $\bar{c}$ is $45^{\circ}$.

To determine $\bar{c}$'s magnitude, apply the dot product definition: $\bar{b} \cdot \bar{c} = |\bar{b}| |\bar{c}| \cos \theta$.
Substitute known values: $8 = 2 |\bar{c}| \cdot \frac{1}{\sqrt{2}}$, leading to $|\bar{c}| = 4\sqrt{2}$.

Since the vectors are coplanar, $\bar{a}$ lies in the plane formed by $\bar{b}$ and $\bar{c}$, while $\bar{b} \times \bar{c}$ is perpendicular to this plane.
Thus, the angle between $\bar{a}$ and $\bar{b} \times \bar{c}$ is $90^{\circ}$.

The magnitude of the cross product is evaluated as:
$|\bar{a} \times (\bar{b} \times \bar{c})| = |\bar{a}| \cdot |\bar{b} \times \bar{c}| \cdot \sin 90^{\circ} = |\bar{b} \times \bar{c}|$.

Compute $|\bar{b} \times \bar{c}| = |\bar{b}| \cdot |\bar{c}| \cdot \sin 45^{\circ} = 2 \cdot 4\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 8$.

Therefore, $|\bar{a} \times (\bar{b} \times \bar{c})| = 8$.