Which expression is equivalent to \( \frac{45 m^{-6} p^{2} v^{12}}{15 m^{-2} p^{8} v^{-4}} \) for all values of \( m, p \), and \( v \) where the expression is defined? A. \( \frac{3 v^{8}}{m^{8} p^{6}} \) B. \( \frac{3 v^{16}}{m^{4} p^{6}} \) C. \( \frac{30 m^{3}}{p^{4} v^{3}} \) D. \( \frac{30 v^{3}}{m^{3} p^{4}} \)

To simplify the expression \( \frac{45 m^{-6} p^{2} v^{12}}{15 m^{-2} p^{8} v^{-4}} \), we start by simplifying the coefficients: \[ \frac{45}{15} = 3 \] Next, we simplify the variables using the laws of exponents: 1. For \( m \): \[ m^{-6} \div m^{-2} = m^{-6 - (-2)} = m^{-6 + 2} = m^{-4} \] 2. For \( p \): \[ p^{2} \div p^{8} = p^{2 - 8} = p^{-6} \] 3. For \( v \): \[ v^{12} \div v^{-4} = v^{12 - (-4)} = v^{12 + 4} = v^{16} \] Putting it all together, we have: \[ \frac{45 m^{-6} p^{2} v^{12}}{15 m^{-2} p^{8} v^{-4}} = 3 m^{-4} p^{-6} v^{16} \] This can be rewritten as: \[ \frac{3 v^{16}}{m^{4} p^{6}} \] Thus, the expression simplifies to \( \frac{3 v^{16}}{m^{4} p^{6}} \), which corresponds to option B.