Question

Find the cube roots of the following numbers by successive subtraction of numbers:
1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, ...
(i) 64
(ii) 512
(iii) 1728

Answer 1

(i)
We have:

$$\begin{gathered} 64 \\ \underline{1} \\ 63 \\ \underline{7} \\ 56 \\ \underline{19} \\ 37 \\ \underline{37} \\ 0 \end{gathered}$$

$\because$ Subtraction is performed 4 times.

$\therefore$ $\sqrt[3]{64}=4$

(ii)
We have:

$$\begin{gathered} 512 \\ \underline{1} \\ 511 \\ \underline{7} \\ 504 \\ \underline{19} \\ 485 \\ \underline{37} \\ 448 \\ \underline{61} \\ 387 \\ \underline{91} \\ 296 \\ \underline{127} \\ 169 \\ \underline{169} \\ 0 \end{gathered}$$
$\because$ Subtraction is performed 8 times.

$\therefore$ $\sqrt[3]{512}=8$

(iii)
We have:

$$\begin{gathered} 1728 \\ \underline{1} \\ 1727 \\ \underline{7} \\ 1720 \\ \underline{19} \\ 1701 \\ \underline{37} \\ 1664 \\ \underline{61} \\ 1603 \\ \underline{91} \\ 1512 \\ \underline{127} \\ 1385 \\ \underline{169} \\ 1216 \\ \underline{217} \\ 999 \\ \underline{271} \\ 728 \\ \underline{331} \\ 397 \\ \underline{397} \\ 0 \end{gathered}$$

$\because$ Subtraction is performed 12 times.

$\therefore$ $\sqrt[3]{1728}=12$