We are given two natural numbers, say $x$ and $y$, with the following conditions:

1. The sum of the two numbers is $x + y = 32$.
2. The sum of their cubes is $x^3 + y^3 = 9728$.

We are tasked with finding the product $x \cdot y$.

Step-by-step Solution:

We know that the identity for the sum of cubes is: $x^3 + y^3 = (x + y)((x + y)^2 - 3xy)$ Substitute $x + y = 32$ into the identity: $x^3 + y^3 = 32(32^2 - 3xy)$ Simplify: $9728 = 32(1024 - 3xy)$ Now, divide both sides by 32: $9728 \div 32 = 1024 - 3xy$ $304 = 1024 - 3xy$ Rearrange to solve for $xy$: $3xy = 1024 - 304$ $3xy = 720$ $xy = \frac{720}{3} = 240$

Thus, the product of the two numbers is $\boxed{240}$.

Would you like more details on any step or have further questions? Here are some related problems:

1. How do you solve for two numbers given their sum and product?
2. Can you factor the sum and difference of cubes?
3. What other identities can be useful for solving cubic equations?
4. How does the relationship between roots and coefficients work in quadratic equations?
5. How can the method of substitution help in solving systems of equations?

Tip: When working with identities, it’s useful to remember common formulas like the sum or difference of cubes, as they simplify complex equations quickly.