Using prime factorization method, find the cube root of
(a) $512$
(b) $2197$

Hint: The cube root of given numbers is to be calculated by prime factorization method. we have to first find the prime factors of the given numbers. Then we have to make the group of the same three numbers and to find the cube root we have to multiply a number from each group. And we will get the required cube root of the given numbers.

Complete step-by-step answer:
(a) Here, the given number is $512$.
To find its cube root firstly we have to find its prime factor so, the prime factor of
$512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
Now make the group of three of the same numbers. This implies
$512 = \left( {2 \times 2 \times 2} \right) \times \left( {2 \times 2 \times 2} \right) \times \left( {2 \times 2 \times 2} \right)$
Now, take one number from each group and multiply them to get its cube root. This implies
$\sqrt[3]{{512}} = 2 \times 2 \times 2$
$\therefore \sqrt[3]{{512}} = 8$
Thus, cube root of $512$ is $8$.
(b) Here, the given number is $2197$.
To find its cube root firstly we have to find its prime factor so, the prime factor of
$2197 = 13 \times 13 \times 13$
Now make the group of three of the same numbers. This implies
$2197 = \left( {13 \times 13 \times 13} \right)$
Now, take one number from each group and multiply them to get its cube root. This implies
$\sqrt[3]{{2197}} = 13$
$\therefore \sqrt[3]{{2197}} = 13$
Thus, cube root of $2197$ is $13$.

Note: Similarly, this prime factorization method is also used to find the square root of the given numbers. To find the square root, firstly find the prime factors of the number and then instead of making a group of three numbers as in above case, make the group of two numbers, then by multiplying a number from each group we will get the square root of the given numbers.