Theory:

Steps to find the cube root of a number through prime factorisation:
Step 1: Find the prime factorisation of the given number.
 
Step 2: Group the factors in pair of three numbers (triplet).
 
Step 3: If there are no factor leftover, then the given number is a perfect cube. Otherwise, it is not a perfect cube.
 
Step 4: Now, take one factor common from each pair and multiply them.
 
Step 5: The obtained product is a cube root of a given number.
Example:
1. Find the value of \(\sqrt[3]{216}\).
 
Solution:
 
Let us first find the prime factor of \(216\).
 
2|216¯2|108¯2|54¯3|27¯3|9¯3|3¯|1
 
Group the factors in pair of three numbers.
 
\(216 = (2 \times 2 \times 2) \times (3 \times 3 \times 3)\)
 
Here, no factor is leftover. Therefore, \(216\) is a perfect cube.
 
Now, take one factor common from each pair and multiply them.
 
\(\sqrt[3]{216} = 2 \times 3 = 6\)
 
Therefore, the value of \(\sqrt[3]{216} = 6\).
 
 
2. Evaluate the value of 219710003.
 
Solution:
 
The given number is 219710003.
 
Let us first factorise the numbers \(2197\) and \(1000\).
 
219710003 \(=\) 2197310003
 
\(=\) 13×13×13310×10×103
 
\(=\) 13331033
 
\(=\) 1310 (Cube and cube root get cancelled)
 
Therefore, the value of 219710003 \(=\) 1310.