1. \( 2 n^{4} m^{3} \cdot 19 n^{2} m^{9} \) \( \square \)

To simplify \( 2 n^{4} m^{3} \cdot 19 n^{2} m^{9} \), you need to multiply the coefficients and then add the exponents of the like bases. The coefficients are \( 2 \) and \( 19 \), so multiply them to get \( 38 \). For the \( n \) terms, add the exponents \( 4 + 2 = 6 \), and for the \( m \) terms, add \( 3 + 9 = 12 \). Thus, the simplified expression is \( 38 n^{6} m^{12} \). In algebra, understanding how to combine like terms and manipulate exponents is crucial for simplifying expressions efficiently. This skill not only helps in examinations but also in various fields such as engineering, economics, and computer science, where mathematical modeling is essential.