Question

You are told that 1331 is a perfect cube. Can you guess without factorization what is its cube root? Similarly, guess the cube roots of 4913, 12167, 32768.

Answer 1

Hint: Divide the given number into 2 groups. Thus find the number in the one’s place and ten’s place by comparing the unit digit of both the groups.

Complete step-by-step answer:
Now let us consider the first given number 1331. We are told that it is a perfect group. Now let us divide 1331 into 2 groups. 1 will be the first group and 1331 in \[{{2}^{nd}}\] group.

Now considering 1 of digit 331, it will be in the ten’s place i.e. \[{{1}^{3}}=1\] and

\[{{2}^{3}}=8\], so 1 lies between 0 and 8. The smaller number among 1 and 2 are 1. So 1

comes in ten’s place.

Now let us consider \[{{2}^{nd}}\] group 1, the digit 1 is in one's place.

Thus the one’s place of 1 is 1 and ten’s place of cube root of 1331 is 1.

\[\therefore \sqrt[3]{1331}=11\].

Now let us find the cube root of 4913.

Let us divide 4913 into 2 groups, 4 and 913.

For 913, the digit 3 is in one's place. 3 comes at a unit place of a number only when its cube root ends in 7. So one’s place of the required cube root is 7.

For another group, \[{{1}^{3}}=1\] and \[{{2}^{3}}=8\]. So 4 lies in between 1 and 8. The

smaller number 1, so the one’s place 1 and tens place of cube root is 7.

\[\therefore \sqrt[3]{4913}=17\]

Now for 12167, can divide into 12 and 167.

For first group 167, the digit 7 is on the one’s place. 7 comes at a unit place of a number only

when its cube root ends with 3. So, at one’s place the required cube root is 3.

For 12, \[{{2}^{3}}=8\] and \[{{3}^{3}}=27\]. So 12 lies between 8 and 27. 2 is the smaller

number. So the ten’s place will be 2 of the cube root.

\[\sqrt[3]{12167}=23\]

For 32768, we can divide it into 768 and 32.

In 768, 8 is at the one’s place, 8 comes at the unit place of a, number when its cube root

ends in 2. So one’s place of the required cube root is 2.

For 32, it lies between \[{{3}^{3}}=27\] and \[{{4}^{3}}=64\]i.e. 32 lies between 27 and 64.

Hence 3 is the smaller number so at ten’s place of cube root is 3.

\[\sqrt[3]{32768}=32\]

Thus we got the required cube roots,

\[\sqrt[3]{1331}=11\], \[\sqrt[3]{4913}=17\], \[\sqrt[3]{12167}=23\], \[\sqrt[3]{32768}=32\].


Note: We can check if the cube is correct by factorization.
\[\begin{align}
  & 1331=11\times 11\times 11\times 1 \\
 & 4913=17\times 17\times 17\times 1 \\
 & 12167=23\times 23\times 23\times 1 \\
 & 32768={{2}^{5}}\times {{2}^{5}}\times {{2}^{5}}=32\times 32\times 32 \\
\end{align}\]