22) \( \frac{4\left(m n^{-1}\right.}{2 m^{2} n^{-2} \cdot 3 m^{3} n^{2} \cdot 4 n m^{2}} \)

To simplify the expression \( \frac{4\left(m n^{-1}\right)}{2 m^{2} n^{-2} \cdot 3 m^{3} n^{2} \cdot 4 n m^{2}} \), let's break it down step by step. First, simplify the denominator: - The denominator is \( 2 \cdot 3 \cdot 4 \cdot m^{2} \cdot m^{3} \cdot m^{2} \cdot n^{-2} \cdot n^{2} \cdot n \). - Calculate \( 2 \cdot 3 \cdot 4 = 24 \). - Combine the like terms for \( m \) and \( n \): \( m^{2+3+2} = m^{7} \) and \( n^{-2+2+1} = n^{1} \). So the denominator simplifies to \( 24 m^{7} n^{1} \). Now, rewrite the fraction: \[ \frac{4 m n^{-1}}{24 m^{7} n^{1}}. \] This can be simplified further: - For the coefficients: \( \frac{4}{24} = \frac{1}{6} \). - For \( m \): \( \frac{m}{m^{7}} = m^{1-7} = m^{-6} \). - For \( n \): \( \frac{n^{-1}}{n^{1}} = n^{-1-1} = n^{-2} \). Now, combining these: \[ \frac{1}{6} m^{-6} n^{-2} = \frac{1}{6 m^{6} n^{2}}. \] Thus, the simplified expression is: \[ \frac{1}{6 m^{6} n^{2}}. \] Happy simplifying!