Explanation:

- Let the two natural numbers be $a$ and $b$.

- The cube of the difference is given: $(a-b{)}^3=1728$.

- The cube root of 1728 is 12. So, $a-b=12$.

- The product of the numbers is given: $ab=108$.

- Use $a=b+12$ and substitute in the product equation: $(b+12)b=108$.

- Solving $b^2+12b-108=0$ gives roots as $b=6$ and $b=-18$.

- Since $b$ is natural, $b=6$, so $a=18$.

- To find the difference between cubes: $a^3-b^3={18}^3-6^3$.

- Calculate the cubes: ${18}^3=5832$, $6^3=216$.

- Difference: $5832-216=5616$.

- The correct answer is Option 2: 5616.

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