(i) 8.65

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

We know that value of \(\sqrt[3]{860} = 9.510\) and \(\sqrt[3]{870} = 9.546\)

So by unitary method, 

∵ For difference (870 – 860 = 10 ) difference in cube root values = 9.546 – 9.510 = 0.036 

∴ For difference (865 – 860 = 5) difference in cube root values 

= \(\frac{0.036}{10}\times5 = 0.018\)

= \(\sqrt[3]{8.65}\) = 9.510 + 0.018 = 9.528

We also have,\(\sqrt[3]{100} = 4.642\) (from table)

∴ \(\sqrt[3]{8.65}\) = \(\frac{\sqrt[3]{865}}{\sqrt[3]{100}}\) = \(\frac{9.528}{4.642} = 2.053.\)

Thus the required cube root is = 2.053.

(ii) 7532

= \(\sqrt[3]{7532}\)

We know that value of \(\sqrt[3]{7532}\) will lie between \(\sqrt[3]{7500}\) and \(\sqrt[3]{7600.}\)

From cube root table we get,

= \(\sqrt[3]{7500}\) = 19.57 and \(\sqrt[3]{7600.}\) = 19.66

So by unitary method,

∵ For difference (7600 – 7500 = 100 ) difference in cube root values = 19.66 – 19.57 = 0.09

∴ For difference (7532 – 7500 = 32) difference in cube root values 

= \(\frac{0.09}{100}\times32 = 0.029\)

= \(\sqrt[3]{7532}\) = \(19.57 + 0.029 = 19.599.\)

Thus the required cube root is = 19.599.

(iii) 833

= \(\sqrt[3]{833}\)

We know that value of \(\sqrt[3]{833}\) will lie between \(\sqrt[3]{830} = \sqrt[3]{840.}\)

From cube root table we get,

= \(\sqrt[3]{830}\) = 9.398 and \( \sqrt[3]{840} = 9.435\)

So by unitary method,

∵ For difference (840 – 830 = 10 ) difference in cube root values = 9.435 – 9.398 = 0.037 

∴ For difference (833 – 830 = 3) difference in cube root values 

= \(\frac{0.037}{10}\times3 = 0.011\)

= \(\sqrt[3]{833}\) = 9.398 + 0.011 = 9.409

(iv) 34.2

= \(\sqrt[3]{340} = \) \(\sqrt[3]{\frac}{342}{10}\) = \(\frac{\sqrt[3]{342}}{\sqrt[3]{10}}\)

We know that value of \(\sqrt[3]{342}\) will lie between \(\sqrt[3]{340} \) and \(\sqrt[3]{350.}\)

From cube root table we get,

= \(\sqrt[3]{340} = \) 6.980 and \(\sqrt[3]{350} = 7.047\)

So by unitary method, 

∵ For difference (350 – 340 = 10 ) difference in cube root values = 7.047 – 6.980 = 0.067 

∴ For difference (342 – 340 = 2) difference in cube root values 

=\(\frac{0.067}{10}\times2 = 0.013\)

= \(\sqrt[3]{342}\) = 6.980 + 0.013 = 6.993

We also have, \(\sqrt[3]{10} = 2.154\) (from table)

∴ \(\sqrt[3]{34.2}\) = \(\frac{\sqrt[3]{342}}{\sqrt[3]{10}}\) = \(\frac{6.993}{2.154} = 3.246.\)

Thus the required cube root is = 3.246.