Proving that: $9.9998\lt \frac{\pi^9}{e^8}\lt 10$?

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?

My approach can be summarized as follows:

1. By a well-known fact about Euler numbers, $\pi^9$ is given by a rational multiple of the fast-converging series $\sum_{n\geq 0}\frac{(-1)^n}{(2n+1)^9}$;
2. By the generalized continued fraction for the hyperbolic cotangent function, it is pretty simple to produce accurate rational approximations of $e^8$;
3. The given inequality can be proved by producing accurate rational approximations for both $\pi^9$ and $e^8$ through the previous points, then by comparing them.

1. $$\pi^9 = \frac{203325460470370265464832}{6820919298826171875}\pm 7\cdot 10^{-5} $$
2. $$ e^8 = \frac{6456755}{2166}\pm 4\cdot 10^{-8}$$
3. Done.