I would like to use mathematical tools to prove that $$9.9998\lt \frac{\pi^9}{e^8}\lt 10$$
With an on-line calculator I got
$$ \frac{\pi^9}{e^8}\approx 9.9998387978$$ But I do not know any good way to prove this. I failed to use Taylor expansion $$e^8 =\sum_{n=0}^{\infty}\frac{8^n}{n!}$$
Any idea?