What is a simple way of computing the following fraction?

Question

Compute the value of the expression:

$$\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}$$

Answer 1

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 $$\begin{align*} &\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)} \\ &\qquad = \frac{(52+6)^2+6(52+6)+18}{4^2-6(4)+18} = \frac{3730}{10}=373 \end{align*}$$