Making use of the cube root table,

The number $0.27$ can be written as $\frac{27}{100}$.

now

$\sqrt[3]{0.27}=\sqrt[3]{\frac{27}{100}}=\frac{\sqrt[3]{27}}{\sqrt[3]{100}}=\frac{3}{\sqrt[3]{100}}$

By cube root table, we have: 

$\sqrt[3]{100}=4.642$

$\therefore \sqrt[3]{0.27}=\frac{3}{\sqrt[3]{100}}=\frac{3}{4.642}=0.646$

Thus, the required cube root is 0.646.