Identifying fifth powers, you find that $\displaystyle{\sqrt[{{5}}]{{{512}{{a}}^{{9}}{{b}}^{{5}}{{c}}^{{14}}}}}={\sqrt[{{5}}]{{{{2}}^{{5}}\cdot{16}\cdot{{a}}^{{5}}\cdot{{a}}^{{4}}\cdot{{b}}^{{5}}\cdot{{\left({{c}}^{{2}}\right)}}^{{5}}\cdot{{c}}^{{4}}}}}\usepackage{cmbright}\Huge\$ . This simplifies to $\displaystyle{2}{a}{b}{{c}}^{{2}}{\sqrt[{{5}}]{{{16}{{a}}^{{4}}{{c}}^{{4}}}}}\usepackage{cmbright}\Huge\$ .