which one is larger? 8 factorial to the power of 1/8 or 9 factorial to the power of 1/9

Question

Which one is larger?

8 factorial to the power of 1/8 or 9 factorial to the power of 1/9. Show your work without calculator.

Answer 1

Hint: Compare $(8!)^9=(8!)(8!)^8$ and $(9!)^8=9^8(8!)^8$.

Answer 2

Define the function $f(x) = x^{\frac{1}{x}}$ and calculate its derivative. You should get $f'(x)=-x^{\frac{1}{x}-2}(ln(x)-1)$. You see that that $\forall x > e: f'(x) < 0$ since $ln(e)=1$. Thus, the function $f(x)$ is strictly decreasing function for $x>e \approx 2.718$. Hence, $f(8) > f(9)$.