implify: \( \frac{32 a b c}{64 a c} \times \frac{8 b c}{4 b^{2}} \)

To simplify the expression, start with the two fractions: \[ \frac{32 a b c}{64 a c} \times \frac{8 b c}{4 b^{2}} \] First, simplify each fraction separately: 1. **For the first fraction:** \[ \frac{32 a b c}{64 a c} = \frac{32 \cancel{a} \cancel{c} b}{64 \cancel{a} \cancel{c}} = \frac{32 b}{64} = \frac{1}{2} b \] 2. **For the second fraction:** \[ \frac{8 b c}{4 b^{2}} = \frac{8 \cancel{b} c}{4 \cancel{b} b} = \frac{8 c}{4 b} = 2 \frac{c}{b} \] Now, multiply the simplified results together: \[ \left(\frac{1}{2} b\right) \times \left(2 \frac{c}{b}\right) = \frac{1}{2} b \cdot 2 \frac{c}{b} = \frac{2bc}{2b} = c \] Thus, the simplified expression is: \[ \boxed{c} \]