(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.