(i) 37800
= \(\sqrt[3]{37800} = \sqrt[3]{2\times2\times2\times3\times3\times3\times175}\) = \(\sqrt[3]{2^3\times3^3\times175}\)
= \(6\times\sqrt[3]{175}\)
We know that value of \(\sqrt[3]{175}\) will lie between \(\sqrt[3]{170}\) and \(\sqrt[3]{180.}\)
From cube root table we get,
= \(\sqrt[3]{170}\) = 5.540 and \(\sqrt[3]{180} = 5.646\)
So by unitary method,
∵ For difference (180 – 170 = 10 ) difference in cube root values = 5.646 – 5.540 = 0.106
∴ For difference (175 – 170 = 5) difference in cube root values
= \(\frac{0.106}{10}\times5 = 0.053\)
= \(\sqrt[3]{175}\) = 5.540 + 0.053 = 5.593
Hence,
= \(\sqrt[3]{37800}= 6\times\sqrt[3]{175} = 6\times5.593 = 5.593\)
Thus, the required cube root is 33.558.
(ii) 0.27
= \(\sqrt[3]{0.27} = \) \(\sqrt[3]{\frac}{27}{100}\) = \(\frac{\sqrt[3]{27}}{\sqrt[3]{100}}\)
From cube root table we get,
= \(\sqrt[3]{27} = 3 \) and \(\sqrt[3]{100}= 4.642\)
Hence,
= \(\sqrt[3]{0.27} = \) \(\frac{\sqrt[3]{27}}{\sqrt[3]{100}}\) = \(\frac{3}{4.642}= 0.646.\)
Thus the required cube root is = 0.646.
(iii) 8.6
= \(\sqrt[3]{8.6}\) = \(\sqrt[3]{\frac}{86}{10}\) = \(\frac{\sqrt[3]{86}}{\sqrt[3]{10}}\)
From cube root table we get,
= \(\sqrt[3]{86} = 4.414 \) and \(\sqrt[3]{10} = 2.154\)
Hence,
= \(\sqrt[3]{86} = \) \(\frac{\sqrt[3]{86}}{\sqrt[3]{10}}\) = \(\frac{4.414}{2.514} = 2.049.\)
Thus the required cube root is = 2.049.
(iv) 0.86
= \(\sqrt[3]{0.86} = \) \(\sqrt[3]{\frac}{86}{100}\) = \(\frac{\sqrt[3]{86}}{\sqrt[3]{100}}\)
From cube root table we get,
= \(\sqrt[3]{86} = 4.414\) and \(\sqrt[3]{100}= 4.642\)
Hence,
= \(\sqrt[3]{0.86} = \) \(\frac{\sqrt[3]{86}}{\sqrt[3]{100}}\) \(\frac{4.414}{4.642} = 0.951.\)
Thus the required cube root is = 0.951.