Which expressions is equivalent to \( \left(\frac{\frac{1}{2} m\left(4 d^{2} m^{3}\right)}{8 d^{4} m^{-2} g^{4}}\right)^{-2} \) ?

To simplify the expression \( \left(\frac{\frac{1}{2} m\left(4 d^{2} m^{3}\right)}{8 d^{4} m^{-2} g^{4}}\right)^{-2} \), we will first simplify the fraction inside the parentheses. 1. **Simplify the numerator:** \[ \frac{1}{2} m (4 d^2 m^3) = \frac{1}{2} \cdot 4 \cdot m \cdot d^2 \cdot m^3 = 2 m^{1+3} d^2 = 2 m^4 d^2 \] 2. **Simplify the denominator:** \[ 8 d^4 m^{-2} g^4 \] 3. **Combine the numerator and denominator:** \[ \frac{2 m^4 d^2}{8 d^4 m^{-2} g^4} \] 4. **Simplify the fraction:** - The coefficient simplifies as follows: \[ \frac{2}{8} = \frac{1}{4} \] - For the \(d\) terms: \[ \frac{d^2}{d^4} = d^{2-4} = d^{-2} \] - For the \(m\) terms: \[ \frac{m^4}{m^{-2}} = m^{4 - (-2)} = m^{4 + 2} = m^6 \] Putting it all together, we have: \[ \frac{2 m^4 d^2}{8 d^4 m^{-2} g^4} = \frac{1}{4} m^6 d^{-2} g^{-4} \] 5. **Now, we need to raise this to the power of \(-2\):** \[ \left(\frac{1}{4} m^6 d^{-2} g^{-4}\right)^{-2} \] 6. **Apply the exponent to each part:** - The coefficient: \[ \left(\frac{1}{4}\right)^{-2} = 4^2 = 16 \] - The \(m\) term: \[ (m^6)^{-2} = m^{-12} \] - The \(d\) term: \[ (d^{-2})^{-2} = d^{4} \] - The \(g\) term: \[ (g^{-4})^{-2} = g^{8} \] 7. **Combine all parts:** \[ 16 m^{-12} d^{4} g^{8} \] Thus, the expression \( \left(\frac{\frac{1}{2} m\left(4 d^{2} m^{3}\right)}{8 d^{4} m^{-2} g^{4}}\right)^{-2} \) simplifies to: \[ \boxed{16 d^4 g^8 m^{-12}} \]