Question

You are told that 1,331 is a perfect cube. Can you guess without factorization what its cube root is? Similarly, guess the cube roots of:​
(i) $1331$
(ii) $4913$
(iii) $12167$
(iv) $32768$

Answer 1

(i) Step 1: Make group digit of 3 digit, starting from right.​

Step 2: First group, i.e., 331 will give you the one’s (or unit’s) digit of the required cube root.​
The first group 331 will give us unit’s digit of cube root of 1331.​
Since 331 ends with 1, so we will see the number whose cube ends with 1.​
$1^3=1$ ​
${11}^3=1331$​
So, unit’s digit of cube of a number will be 1 only when unit’s digit of the number is 1.​

∴ Unit’s digit of cube root of 1331 = 1
Step 3: Second group, i.e., 1 will give you the ten’s digit of the required cube root.​

Now, for second group.​
1​
Second group will give us ten’s digit of the cube root of 1331.​
We will find the number whose cube is 1.​
We note that ​
$1^3=1$​
​So, ten’s place of cube root of 1331 will be 1.​
Therefore, cube root of 1331 = 11​

(ii) Step 1: Make group of 3 digit, starting from right.​

Step 2: First group, i.e., 913 will give you the one’s (or unit’s) digit of the required cube root.​

The first group 913 will give us unit's digit of cube root of 4913.
Since 913 ends with 3, so we will see the number whose cube ends with 3.
$7^3=343$
${17}^3=4913$
So, unit's digit of cube of a number will be 3 only when unit's digit of the number is 7.
$\therefore$ Unit's digit of cube root of $4913=7$
Step 3: Second group, i.e., 4 will give you the ten’s digit of the required cube root.​

Now for second group.
4
Second group will give us ten's digit of the cube root of 4913.
We will find the number whose cubes are less than and greater than 4 and the unit's digit of smaller number will be the ten's digit of cube root of 4913.
We note that
$1^3=3\text{\&}{2}^3=8$
So, $1^3<4<2^3$
$\therefore$ So, unit's digit of smaller number = 1
Therefore, ten's place of cube root of 4913 = 1
Therefore, cube root of 4913 = 17

(iii) Step 1: Make group of 3 digit, starting from right.​

Step 2: First group, i.e., 167 will give you the one’s (or unit’s) digit of the required cube root.​

The first group 167 will give us unit's digit of cube root of 12167.
Since 167 ends with 7, so we will see the number whose cube ends with 7.
$3^3=27$
${13}^3=2797$
So, unit's digit of cube of a number will be 7 only when unit's digit of the number is 3.
$\therefore$ Unit's digit of cube root of 12167 = 3
Step 3: Second group, i.e., 12 will give you the ten’s digit of the required cube root.​

Now, for second group,
12
Second group will give us ten's digit of the cube root of 12167.
We will find the numbers whose cubes are less than and greater than 12 and the unit's digit of smaller number will be the ten's digit of cube root of 12167.
We note that
$2^3=8\text{\&}{3}^3=27$
So, $2^3<12<3^3$
$\therefore$ So, unit's digit of smaller number = 2
Therefore, ten's place of cube root of 12167 = 2
Therefore, cube root of 12167 = 23

(iv) Step 1: Make group of 3 digit, starting from right.​

Step 2: First group, i.e., 768 will give you the one’s (or unit’s) digit of the required cube root.​

The first group 768 will give us unit's digit of cube root of 32768.
Since 768 ends with 8, so we will see the number whose cube ends with 8.
$2^3=8$
${12}^3=1728$
So, unit's digit of cube of a number will be 8 only when unit's digit of the number is 2.
$\therefore$ Unit's digit of cube root of 32768 = 2
Step 3: Second group, i.e., 32 will give you the ten’s digit of the required cube root.​

Now, for second group.
32
Second group will give us ten's digit of the cube root of 32768.
We will find the numbers whose cubes are less than and greater than 32 and the unit's digit of smaller number will be the ten's digit of cube root of 32768.
We note that
$3^3=27\text{\&}{4}^3=64$
So, $3^3<32<4^3$
$\therefore$ So, unit's digit of smaller number = 3
Therefore, ten's place of cube root of 12167 = 3
Therefore, cube root of 32768 = 32