Identifying cubes under the radical, you find $\displaystyle{\sqrt[{{3}}]{{{512}{{a}}^{{9}}{{b}}^{{5}}{{c}}^{{14}}}}}={\sqrt[{{3}}]{{{{8}}^{{3}}\cdot{{\left({{a}}^{{3}}\right)}}^{{3}}\cdot{{b}}^{{3}}\cdot{{b}}^{{2}}\cdot{{\left({{c}}^{{4}}\right)}}^{{3}}\cdot{{c}}^{{2}}}}}\usepackage{cmbright}\Huge\$ . This simplifies to $\displaystyle{8}{{a}}^{{3}}{b}{{c}}^{{4}}{\sqrt[{{3}}]{{{{b}}^{{2}}{{c}}^{{2}}}}}\usepackage{cmbright}\Huge\$ .