<s>[INST] What is a simple way of computing the following fraction? Compute the value of the expression:
$$\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}$$
 [/INST] Note that
$$x^4+324=(x^2-6x+18)(x^2+6x+18).$$
Hence
$$
\begin{align*}
\frac{(n+6)^4+324}{n^4+324}
&= \frac{((n+6)^2-6(n+6)+18)((n+6)^2+6(n+6)+18)}{(n^2-6n+18)(n^2+6n+18)} \\
&= \frac{(n^2+6n+18)((n+6)^2+6(n+6)+18)}{(n^2-6n+18)(n^2+6n+18)} \\
&=\dfrac{(n+6)^2+6(n+6)+18}{n^2-6n+18}
\end{align*}$$
which means that most of the terms cancel and we are left with
$$\begin{align*}
&\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)} \\
&\qquad = \frac{(52+6)^2+6(52+6)+18}{4^2-6(4)+18} = \frac{3730}{10}=373
\end{align*}$$
</s>