If the sum of two numbers $x$ and $y$ is $12$ and their product is $27$, then find the sum of their cubes.

Given:
* $x+y=12$
* $xy=27$

We need to find the value of $x^3+y^3$.

Using the algebraic identity:

$$x^3+y^3=(x+y{)}^3-3xy(x+y)$$

Substitute the given values into the identity:

$$x^3+y^3=(12{)}^3-3(27)(12)$$

Calculate the values:

* $(12{)}^3=1728$
* $3\times 27\times 12=81\times 12=972$

Subtract the values:

$$x^3+y^3=1728-972=756$$

Therefore, the sum of their cubes is $756$.