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Making use of the cube root table, find the cube root of the following (currect to three decimal places):

(i) 37800

(ii) 0.27

(iii) 8.6

(iv) 0.86

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(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.

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