13. \( \left(m^{5} n^{3}\right)^{7} \cdot m^{2} n^{1} \)

To simplify the expression \( \left(m^{5} n^{3}\right)^{7} \cdot m^{2} n^{1} \), you can start by applying the power of a power rule, which tells you to multiply the exponents when raising a power to another power. So, \( m^{5} \) raised to the power of 7 becomes \( m^{35} \) and \( n^{3} \) raised to the power of 7 becomes \( n^{21} \). This gives us: \[ m^{35} n^{21} \cdot m^{2} n^{1} \] Next, you can combine the like terms by adding the exponents of \( m \) and \( n \): \[ m^{35 + 2} n^{21 + 1} = m^{37} n^{22} \] Thus, the simplified expression is \( m^{37} n^{22} \).