Question

Find the smallest number that must be subtracted from those of the numbers in question 2 which are not perfect cubes, to make them perfect cubes. What are the corresponding cube roots?

Answer 1

(i)
We have:

$$\begin{gathered} 130 \\ \underline{1} \\ 129 \\ \underline{7} \\ 122 \\ \underline{19} \\ 103 \\ \underline{37} \\ 66 \\ \underline{61} \\ 5 \end{gathered}$$

$\because$ The next number to be subtracted is 91, which is greater than 5.

$\therefore$ 130 is not a perfect cube.

However, if we subtract 5 from 130, we will get 0 on performing successive subtraction and the number will become a perfect cube.

If we subtract 5 from 130, we get 125. Now, find the cube root using successive subtraction.

We have:

$$\begin{gathered} 125 \\ \underline{1} \\ 124 \\ \underline{7} \\ 117 \\ \underline{19} \\ 98 \\ \underline{37} \\ 61 \\ \underline{61} \\ 0 \end{gathered}$$
$\because$ The subtraction is performed 5 times.

$\therefore$ $\sqrt[3]{125}=5$

Thus, it is a perfect cube.

(ii)
We have:

$$\begin{gathered} 345 \\ \underline{1} \\ 344 \\ \underline{7} \\ 337 \\ \underline{19} \\ 318 \\ \underline{37} \\ 281 \\ \underline{61} \\ 220 \\ \underline{91} \\ 129 \\ \underline{127} \\ 2 \end{gathered}$$
$\because$ The next number to be subtracted is 161, which is greater than 2.

$\therefore$ 345 is not a perfect cube.

However, if we subtract 2 from 345, we will get 0 on performing successive subtraction and the number will become a perfect cube.

If we subtract 2 from 345, we get 343. Now, find the cube root using successive subtraction.

$$\begin{gathered} 343 \\ \underline{1} \\ 342 \\ \underline{7} \\ 335 \\ \underline{19} \\ 316 \\ \underline{37} \\ 279 \\ \underline{61} \\ 218 \\ \underline{91} \\ 127 \\ \underline{127} \\ 0 \end{gathered}$$

$\because$ The subtraction is performed 7 times.

$\therefore$ $\sqrt[3]{343}=7$

Thus, it is a perfect cube.

(iii)
We have:

$$\begin{gathered} 792 \\ \underline{1} \\ 791 \\ \underline{7} \\ 784 \\ \underline{19} \\ 765 \\ \underline{37} \\ 728 \\ \underline{61} \\ 667 \\ \underline{91} \\ 576 \\ \underline{127} \\ 449 \\ \underline{169} \\ 280 \\ \underline{217} \\ 63 \end{gathered}$$

$\because$ The next number to be subtracted is 271, which is greater than 63.

$\therefore$ 792 is not a perfect cube.

However, if we subtract 63 from 792, we will get 0 on performing successive subtraction and the number will become a perfect cube.

If we subtract 63 from 792, we get 729. Now, find the cube root using the successive subtraction.

We have:

$$\begin{gathered} 729 \\ \underline{1} \\ 728 \\ \underline{7} \\ 721 \\ \underline{19} \\ 702 \\ \underline{37} \\ 665 \\ \underline{61} \\ 604 \\ \underline{91} \\ 513 \\ \underline{127} \\ 386 \\ \underline{169} \\ 217 \\ \underline{217} \\ 0 \end{gathered}$$

$\because$ The subtraction is performed 9 times.

$\therefore$ $\sqrt[3]{729}=9$

Thus, it is perfect cube.