Which expression is equivalent to \( \frac{\left(3 m^{-1} n^{2}\right)^{4}}{\left(2 m^{-2} n\right)^{3}} \) ? Assume \( m \neq 0, n \neq 0 \)

We start with the expression \[ \frac{\left(3 m^{-1} n^{2}\right)^{4}}{\left(2 m^{-2} n\right)^{3}}. \] **Step 1. Expand the numerator** Raise each factor in the numerator to the power of 4: \[ \left(3 m^{-1} n^{2}\right)^4 = 3^4 \cdot \left(m^{-1}\right)^4 \cdot \left(n^2\right)^4 = 81 m^{-4} n^8. \] **Step 2. Expand the denominator** Similarly, raise each factor in the denominator to the power of 3: \[ \left(2 m^{-2} n\right)^3 = 2^3 \cdot \left(m^{-2}\right)^3 \cdot n^3 = 8 m^{-6} n^3. \] **Step 3. Form the fraction and simplify** Now we have: \[ \frac{81 m^{-4} n^8}{8 m^{-6} n^3}. \] Simplify by combining like bases: - For \( m \): \[ m^{-4} \div m^{-6} = m^{-4 - (-6)} = m^2. \] - For \( n \): \[ n^8 \div n^3 = n^{8-3} = n^5. \] - The numerical coefficients are: \[ \frac{81}{8}. \] Thus, the simplified expression is: \[ \frac{81 m^{2} n^{5}}{8}. \] The equivalent expression is \[ \frac{81 m^{2} n^{5}}{8}. \]