Find the cube root of a given number through estimation: $2197$

Hint: Start with finding the unit digit of the given number by using the property that says that the unit digit of the product of numbers is the same as the unit digit of the product of the unit digits of the numbers. After finding the unit digit, find the tens multiple between which the required number lies, i.e. ${10^3} < 2197 < {20^3}$ .

Complete step-by-step answer:
Here in this problem, we are given a positive integer $2197$ . We need to find out the cube root of this number through estimation without performing a long calculation.
As we know that the unit digit of the product of two numbers is the same as the unit digit of the product of the unit digits of the two numbers, i.e.
Considering two numbers $abcd{\text{ and }}wxyz$ where ‘a’, ‘b’, ‘c’, ‘d’, ‘w’, ‘x’, ‘y’ and ‘z’ are the digits of these two given numbers.
$ \Rightarrow $ Unit digit of $\left( {abcd \times wxyz} \right)$ $ = $ Unit digit of $\left( {d \times z} \right)$
If we consider the given number to be the cube of a number ‘m’, then $2197 = {m^3} = m \times m \times m$
Then from the above relation, we can say:
$ \Rightarrow $ Unit digit of $\left( {2197} \right)$ $ = $ Unit digit of $\left( {m \times m \times m} \right)$ $ = {\text{ Unit digit of}}\left( {{{\left( {{\text{Unit digit of }}\left( m \right)} \right)}^3}} \right)$
As we know that the only single-digit number which has $7$ as the unit digit of its cube is $3$
$ \Rightarrow 3 \times 3 \times 3 = {3^3} = 27 \Rightarrow {\text{ Unit digit of }}\left( {27} \right) = 7$
Therefore, we can conclude that the unit digit of the required number ‘m’ is $3$
Now we can find the number between which the given cube number will lie.
As we know ${10^3} = 10 \times 10 \times 10 = 1000$ and ${20^3} = 20 \times 20 \times 20 = {2^3} \times 1000 = 8000$
But the given number lies between $1000{\text{ and }}8000$
$ \Rightarrow 1000 < 2197 < 8000 \Rightarrow {10^3} < \sqrt[3]{{2197}} < {20^3} \Rightarrow 10 < m < 20$
Thus, we came up to the conclusion that the unit digit of the cube root of $2197$ is $3$ and the cube root of $2197$ lies between $10$ and $20$ .
Therefore, the required cube root of the required number can only be $13$ .
$ \Rightarrow \sqrt[3]{{2197}} = 13$ .

Note: In questions like this, knowing the nearest cubes of the number or finding the cubes of multiples of ten always plays an important role. The cubes or squares of multiples of tens can be easily found by simply multiplying the non-zero together then placing the same number of zeroes from the right. For example, $10 \times 20 \times 30 = 1 \times 2 \times 3 \times 1000 = 6 \times 1000 = 6000$ .