Making use of the cube root table,

By prime factorisation, we have:

$5112=2^{3} \times 3^{2} \times 71 \Rightarrow \sqrt[3]{5112}=2 \times \sqrt[3]{9} \times \sqrt[3]{71}$'

By the cube root table, we have:

$\sqrt[3]{9}=2.080$ and $\sqrt[3]{71}=4.141$

$\therefore \sqrt[3]{5112}=2 \times \sqrt[3]{9} \times \sqrt[3]{71}=2 \times 2.080 \times 4.141=17.227$ (upto three decimal places)

Thus, the required cube root is 17.227.