Question

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

Answer 1

Hint: For solving this problem, we divide the given number into groups of three digits. By considering the individual groups, we try to obtain the cube root of the respective group. After obtaining the cube root, we place the cube root according to its place value. Using this methodology, we can obtain our answer.

Complete Step-by-Step solution:
For the first number 1331, we divide 1331 into groups of three digits from the right. So, two groups are 331 and 1 for 1331.The digit 1 is at one's place for the first group 331. So, by cube root analysis, cube root of 1 is 1. So, one's place of the required cube root is 1.
Considering the second group having 1, ${{1}^{3}}=1\text{ and }{{2}^{3}}=8$. So, 1 lies between 0 and 8 and hence the ten’s place of cube root 1331 is 1.
Hence, the cube root of 1331 is 11 without factorization.
For 4913:
Again, operating in the similar manner, the two groups are 913 and 4. For first group 913, the digit 3 is at one's place. So, by cube root analysis, cube root of 7 is 3. So, one's place of the required cube root is 7.
For another group, ${{1}^{3}}=1\text{ and }{{2}^{3}}=8$. So, 4 lies between 1 and 8. The smaller number among 1 and 2 are 1 and ten's place of cube root 4913 is 1.
Hence, $\sqrt[3]{4913}=17$.
For 12167:
Again, we divide 12167 into two groups 167 and 12 respectively. For first group 167, the digit 7 is at one's place and by cube root analysis, one's place of the required cube root is 3.
For another group, ${{2}^{3}}=8\text{ and }{{3}^{3}}=27$. So, 12 lies between 8 and 27. The smaller number among 2 and 3 are 2.
Hence, $\sqrt[3]{12167}=23$.
For 32768:
Two groups are 768 and 32. For first group 768, the digit 8 is at one's place which indicates the presence of 2 ant unit’s place.
For another group, ${{3}^{3}}\text{=27 and }{{\text{4}}^{3}}=64$. So, 32 lies between 27 and 64. Thus, the ten's place of cube root 32768 is 3.
Hence, $\sqrt[3]{32768}=32$.

Note: The key concept for solving the problem is the knowledge of approximation of the cube root of a number by using the separation method. This method is very useful for calculating the cube root of a large number in little time. This question provides a time saving strategy.