Simplify. \[ \left(4 a^{2}\right)^{2} a^{9}+a^{5} a^{8} \]

To simplify the expression \(\left(4 a^{2}\right)^{2} a^{9}+a^{5} a^{8}\), we can start by simplifying each term separately. First, let's simplify \(\left(4 a^{2}\right)^{2}\): \[ \left(4 a^{2}\right)^{2} = 4^{2} (a^{2})^{2} = 16 a^{4} \] Now, we can multiply this by \(a^{9}\): \[ 16 a^{4} a^{9} = 16 a^{4+9} = 16 a^{13} \] Next, we simplify the second term \(a^{5} a^{8}\): \[ a^{5} a^{8} = a^{5+8} = a^{13} \] Now, we can combine both simplified terms: \[ 16 a^{13} + a^{13} = (16 + 1) a^{13} = 17 a^{13} \] Thus, the simplified expression is: \[ \boxed{17 a^{13}} \]