Does there exist a natural number $n$ such that $\sqrt[n]{17 \sqrt{5}+38}+\sqrt[n]{17 \sqrt{5}-38}=2 \sqrt{5}$?
Note that $(\sqrt{5} \pm 2)^{3}=5 \sqrt{5} \pm 30+12 \sqrt{5} \pm 8=17 \sqrt{5} \pm 38$. Therefore, the given equality is true for $n=3$.

## Answer

It exists.