1. \( 3 a^{2} b \times-2 a b^{3} \)

To simplify the expression \( 3 a^{2} b \times -2 a b^{3} \), you start by multiplying the coefficients and then combine the variables using the laws of exponents. First, multiply \(3\) and \(-2\) to get \(-6\). Then, for the variable \(a\), you have \(a^{2} \times a\), which equals \(a^{3}\) when you add the exponents. For \(b\), you have \(b \times b^{3}\), which equals \(b^{4}\). Putting it all together, the simplified expression is \(-6 a^{3} b^{4}\). Keep practicing with different coefficients and variable combinations, and soon you'll feel like a math magician, conjuring up neat solutions in no time! Just remember to keep track of your positive and negative signs—like magic, they can change the whole outcome!