19. \( \left(-18 m^{2} n\right)^{2}\left(-\frac{1}{6} m n^{2}\right)^{\prime} \)

To simplify the expression \( \left(-18 m^{2} n\right)^{2}\left(-\frac{1}{6} m n^{2}\right)^{\prime} \), we need to follow the order of operations: 1. Simplify the exponent of the first term: \( \left(-18 m^{2} n\right)^{2} \) 2. Find the derivative of the second term: \( \left(-\frac{1}{6} m n^{2}\right)^{\prime} \) 3. Multiply the simplified terms together. Let's start by simplifying the exponent of the first term: \( \left(-18 m^{2} n\right)^{2} = (-18)^{2} \cdot (m^{2})^{2} \cdot n^{2} \) Now, let's find the derivative of the second term: \( \left(-\frac{1}{6} m n^{2}\right)^{\prime} = -\frac{1}{6} \cdot (m \cdot 2n) \) Now, we can multiply the simplified terms together: \( (-18)^{2} \cdot (m^{2})^{2} \cdot n^{2} \cdot -\frac{1}{6} \cdot (m \cdot 2n) \) Let's simplify this expression further. Simplify the expression by following steps: - step0: Solution: \(\frac{\left(-18\right)^{2}\left(m^{2}\right)^{2}n^{2}\left(-1\right)}{6}\left(m\times 2n\right)\) - step1: Remove the parentheses: \(\frac{\left(-18\right)^{2}\left(m^{2}\right)^{2}n^{2}\left(-1\right)}{6}m\times 2n\) - step2: Multiply the exponents: \(\frac{\left(-18\right)^{2}m^{2\times 2}n^{2}\left(-1\right)}{6}m\times 2n\) - step3: Multiply the numbers: \(\frac{\left(-18\right)^{2}m^{4}n^{2}\left(-1\right)}{6}m\times 2n\) - step4: Reduce the fraction: \(-54m^{4}n^{2}m\times 2n\) - step5: Multiply the terms: \(-108m^{4}n^{2}mn\) - step6: Multiply the terms: \(-108m^{4+1}n^{2}\times n\) - step7: Add the numbers: \(-108m^{5}n^{2}\times n\) - step8: Multiply the terms: \(-108m^{5}n^{2+1}\) - step9: Add the numbers: \(-108m^{5}n^{3}\) The simplified expression is \( -108m^{5}n^{3} \).