Making use of the cube root table,

The number $0.86$ could be written as $\frac{86}{100}$.

Now

$\sqrt[3]{0.86}=\sqrt[3]{\frac{86}{100}}=\frac{\sqrt[3]{86}}{\sqrt[3]{100}}$

By cube root table, we have: 

$\sqrt[3]{86}=4.414$ and $\sqrt[3]{100}=4.642$

$\therefore \sqrt[3]{0.86}=\frac{\sqrt[3]{86}}{\sqrt[3]{100}}=\frac{4.414}{4.642}=0.951$ (upto three decimal places)

Thus, the required cube root is 0.951.