Making use of the cube root table,

Solution:

The number $8.65$ could be written as $\frac{865}{100}$.

Now

$\sqrt[3]{8.65}=\sqrt[3]{\frac{865}{100}}=\frac{\sqrt[3]{865}}{\sqrt[3]{100}}$

Also

$860<865<870 \Rightarrow \sqrt[3]{860}<\sqrt[3]{865}<\sqrt[3]{870}$

From the cube root table, we have: 

$\sqrt[3]{860}=9.510$ and $\sqrt[3]{870}=9.546$

For the difference (870-">−-860), i.e., 10, the difference in values

$=9.546-9.510=0.036$

$\therefore$ For the difference of $(865-860)$, i.e., 5 , the difference in values

$=\frac{0.036}{10} \times 5=0.018$ (upto three decimal places)

$\therefore \sqrt[3]{865}=9.510+0.018=9.528$ (upto three decimal places)

From the cube root table, we also have:

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

$\therefore \sqrt[3]{8.65}=\frac{\sqrt[3]{865}}{\sqrt[3]{100}}=\frac{9.528}{4.642}=2.053$ (upto three decimal places)

Thus, the required cube root is 2.053.