Multiply. \[ 8 w^{6} \cdot 2 u \cdot 2 u^{2} w^{7} \]

To multiply the expression \( 8 w^{6} \cdot 2 u \cdot 2 u^{2} w^{7} \), we start by multiplying the coefficients and then combining the like terms. First, we multiply the numerical coefficients: \[ 8 \cdot 2 \cdot 2 = 32 \] Next, we combine the \( u \) terms. We have \( 2 u \) and \( 2 u^{2} \), so: \[ u \cdot u^{2} = u^{3} \] Now, we combine the \( w \) terms. We have \( w^{6} \) and \( w^{7} \): \[ w^{6} \cdot w^{7} = w^{6 + 7} = w^{13} \] Putting it all together, we combine the results we found: \[ 32 u^{3} w^{13} \] Therefore, the final result is: \[ \boxed{32 u^{3} w^{13}} \]