elementary number theory - Prove $(3+sqrt{11})^{1/3}$ is irrational.

I can't say I've gotten very far. You can show $3 + \sqrt{11}$ is irrational, call it $a$. Then I tried supposing it's rational, i.e.:

$a^{1/3}$ = $\frac{m}{n}$ for $m$ and $n$ integers.

You can write $m$ and $n$ in their canonical factorizations, then cube both sides of the equation...but I can't seem to derive a contradiction.

Answer

If $(3 + \sqrt{11})^\frac{1}{3}=\frac{p}{q}$ were a rational, then $3 + \sqrt{11}=\frac{p^3}{q^3}$ would also be a rational.