Which of these numbers is greater: $\sqrt[5]{5}$ or $\sqrt[4]{4}$?

Question

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 $x^5 > y^4$.

Answer 1

$\text{}$$5^4<4^5$$\text{}$

Now, take the 20th root on both sides of the inequality if you can!

Answer 2

If $x_0$ is some positive natural number (or in fact any real number greater than $\tfrac{1}{\text e}$), then $$\left(\frac{\text d}{\text dx}x^x\right)_{x=x_0}=\left(x^x(\ln(x)+1)\right)_{x=x_0}>0.$$ The function is smooth and growing, so bigger numbers $x$ give bigger $x^x$.