Which of these numbers is greater: $\sqrt[5]{5}$ or $\sqrt[4]{4}$? I know that this is the question of elementary mathematics but how to logically check which of these numbers is greater: $\sqrt[5]{5}$ or $\sqrt[4]{4}$?
It seems to me that since number $5$ is greater than $4$ and we denote $\sqrt[5]{5}$ as $x$ and $\sqrt[4]{4}$ as $y$ then $x^5 > y^4$.

The function $$n^{1/n}=e^{(1/n)\log n}$$ goes to $1$ as $n$ becomes infinite. Also, taking derivatives shows that it is monotonically decreasing whenever $n>e.$ If we consider only integers now, we have that for $n=4,5,6,\ldots,$ the sequence decreases.
It follows that $$4^{1/4}>5^{1/5}.$$
