Problem - 3232
Find the value of $x^3+x^2y+xy^2+y^3$ if $x=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ and $y=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$.
There are several ways to solve such problems. One typical approach is shown below:
From the given condition, we find $xy=1$ and $x+y = 8$. Then
$$x^3+x^2y+xy^2+y^3 = x^2(x+y)+y^2(x+y)=(x+y)(x^2+y^2)=(x+y)((x+y)^2-2xy)=8\times(8^2-2\times 1)=\boxed{496}$$