### 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 $$216$$.

$\begin{array}{l}\underset{¯}{2|216\phantom{\rule{0.441em}{0ex}}}\\ \underset{¯}{2|108\phantom{\rule{0.441em}{0ex}}}\\ \underset{¯}{2|54\phantom{\rule{0.882em}{0ex}}}\\ \underset{¯}{3|27\phantom{\rule{0.882em}{0ex}}}\\ \underset{¯}{3|9\phantom{\rule{1.176em}{0ex}}}\\ \underset{¯}{3|3\phantom{\rule{1.176em}{0ex}}}\\ \phantom{\rule{0.441em}{0ex}}|\phantom{\rule{0.147em}{0ex}}1\end{array}$

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 $\sqrt[3]{\frac{2197}{1000}}$.

Solution:

The given number is $\sqrt[3]{\frac{2197}{1000}}$.

Let us first factorise the numbers $$2197$$ and $$1000$$.

$\sqrt[3]{\frac{2197}{1000}}$ $$=$$ $\frac{\sqrt[3]{2197}}{\sqrt[3]{1000}}$

$$=$$ $\frac{\sqrt[3]{13×13×13}}{\sqrt[3]{10×10×10}}$

$$=$$ $\frac{\sqrt[3]{{13}^{3}}}{\sqrt[3]{{10}^{3}}}$

$$=$$ $\frac{13}{10}$ (Cube and cube root get cancelled)

Therefore, the value of $\sqrt[3]{\frac{2197}{1000}}$ $$=$$ $\frac{13}{10}$.