Prove or disprove: For all positive integers $ n$ , $\sqrt[3]{n}+\sqrt[3]{n+1}$ are irrational numbers. Prove or disprove: For all positive integers  $ n$ ,
$\sqrt[3]{n}+\sqrt[3]{n+1}$  are irrational numbers.

Note that $(x+y)^{3} = x^{3} +y^{3} + 3xy(x+y).$ If $n$ or $n+1$ is the cube of an integer, the result is clear by the uniqueness of prime factorization, so we assume that neither is the cube of an integer. Then neither is $n(n+1)$. Set $x = n^{\frac{1}{3}}$ and $y = (n+1)^{\frac{1}{3}}.$ If $x+y$ is rational, then so are $(x+y)^{3}$ and $3xy(x+y)$. Hence $3xy$ is rational. Thus $27n(n+1)$ must be the cube of an integer, a contradiction.
