12. Given that $x$ and $y$ are positive real numbers such that $(x+y)^{2}=2500$ and $x y=500$, find the exact value of $x^{3}+y^{3}$.
12. Answer: 50000
$$
\begin{aligned}
x^{3}+y^{3} & =(x+y)\left(x^{2}-x y+y^{2}\right)=(x+y)\left[(x+y)^{2}-3 x y\right] \\
& =50[2500-3(500)]=50000 .
\end{aligned}
$$