\( d \neq \frac { 8 m ^ { 2 } n } { 9 m ^ { - 3 } n ^ { 2 } } \div \frac { 16 m ^ { 3 } n ^ { - 1 } } { 3 m ^ { 4 } n } \)

We begin with the expression \[ d \neq \frac{8m^2 n}{9m^{-3} n^2} \div \frac{16m^3 n^{-1}}{3m^4 n}. \] Since division is equivalent to multiplying by the reciprocal, we rewrite the expression as \[ d \neq \frac{8m^2 n}{9m^{-3} n^2} \times \frac{3m^4 n}{16m^3 n^{-1}}. \] **Step 1. Simplify the first fraction** In the fraction \[ \frac{8m^2 n}{9m^{-3} n^2}, \] apply the laws of exponents for division: - For \(m\): \(m^2 \div m^{-3} = m^{2 - (-3)} = m^{5}\). - For \(n\): \(n \div n^2 = n^{1-2} = n^{-1}\). Thus the fraction simplifies to \[ \frac{8}{9} \, m^5 n^{-1}. \] **Step 2. Simplify the reciprocal of the second fraction** We take the reciprocal of \[ \frac{16m^3 n^{-1}}{3m^4 n}, \] which gives \[ \frac{3m^4 n}{16m^3 n^{-1}}. \] Now simplify using exponent rules: - For \(m\): \(m^4 \div m^3 = m^{4-3} = m^{1}\). - For \(n\): \(n \div n^{-1} = n^{1-(-1)} = n^{2}\). Thus the reciprocal simplifies to \[ \frac{3}{16} \, m n^2. \] **Step 3. Multiply the simplified expressions** Now multiply the two simplified parts: \[ \left(\frac{8}{9} \, m^5 n^{-1}\right) \times \left(\frac{3}{16} \, m n^2\right). \] **Combine the coefficients:** \[ \frac{8}{9} \times \frac{3}{16} = \frac{24}{144} = \frac{1}{6}. \] **Combine the powers of \(m\):** \[ m^5 \times m = m^{5+1} = m^6. \] **Combine the powers of \(n\):** \[ n^{-1} \times n^2 = n^{-1+2} = n. \] Thus the product is \[ \frac{1}{6} \, m^6 n. \] **Final Result:** \[ d \neq \frac{1}{6} m^6 n. \]