Question

Making use of the cube root table, find the cube root
1346.

Answer 1

By prime factorisation, we have:

\[1346 = 2 \times 673 \Rightarrow \sqrt[3]{1346} = \sqrt[3]{2} \times \sqrt[3]{673}\]

Also

\[670 < 673 < 680 \Rightarrow \sqrt[3]{670} < \sqrt[3]{673} < \sqrt[3]{680}\]

From the cube root table, we have:

\[\sqrt[3]{670} = 8 . 750 \text{ and }  \sqrt[3]{680} = 8 . 794\]

For the difference (680 - 670), i.e., 10, the difference in the values

\[= 8 . 794 - 8 . 750 = 0 . 044\]

∴  For the difference of (673 - 670), i.e., 3, the difference in the values 

\[= \frac{0 . 044}{10} \times 3 = 0 . 0132 = 0 . 013\]  (upto three decimal places)

∴ \[\sqrt[3]{673} = 8 . 750 + 0 . 013 = 8 . 763\]

Now \[\sqrt[3]{1346} = \sqrt[3]{2} \times \sqrt[3]{673} = 1 . 260 \times 8 . 763 = 11 . 041\] (upto three decimal places)

Thus, the answer is 11.041.