8. Use the product rule to simplify the \[ \left(m^{6} n\right)\left(m^{15} n^{4}\right) \]

To simplify the expression \(\left(m^{6} n\right)\left(m^{15} n^{4}\right)\) using the product rule, you can combine the like bases by adding their exponents. For the \(m\) terms, you add \(6\) and \(15\) to get \(m^{6+15} = m^{21}\). For the \(n\) terms, you add \(1\) and \(4\) to get \(n^{1+4} = n^{5}\). So, putting it all together, you have: \[ \left(m^{6} n\right)\left(m^{15} n^{4}\right) = m^{21} n^{5}. \]