To simplify the expression, find the cube root of 512 and the cube root of $\displaystyle{{p}}^{{12}}\usepackage{cmbright}\Huge\$ .

$\displaystyle{\sqrt[{{3}}]{{{512}{{p}}^{{12}}}}}={\sqrt[{{3}}]{{{8}\cdot{8}\cdot{8}\cdot{{\left({{p}}^{{4}}\right)}}^{{3}}}}}={\sqrt[{{3}}]{{{{8}}^{{3}}\cdot{{\left({{p}}^{{4}}\right)}}^{{3}}}}}={\sqrt[{{3}}]{{{{8}}^{{3}}}}}\cdot{\sqrt[{{3}}]{{{{\left({{p}}^{{4}}\right)}}^{{3}}}}}={8}{{p}}^{{4}}\usepackage{cmbright}\Huge\$