Need help solving a exponent+multiplication problem The question goes as follows:
$$\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}$$
Of course the question tests for knowledge in logic and simplification as who would keep calculating this sum.  I observed that each term here can be represented as $$x^4+324$$I thought for a minute and I realised that you can express $324$ as $$324=18^2=(2\times3^2)^2$$
So substituting that into the equation, I get $$x^4+4\times3^4$$  which reminds me of $x^4+4y^4$. My professor gave me a hint to add $$(2\times x^2\times2y^2)-(2\times x^2\times2y^2)$$ to $$x^4+4y^4$$So I did, and I got $$x^4+4y^4+(2\times x^2\times2y^2)-(2\times x^2\times2y^2)$$
$$=x^4+4y^4+(2\times x^2\times2y^2)-(2\times x^2\times2y^2)$$
$$=x^4+4y^4+(4x^2y^2)-(2\times x^2\times2y^2)$$
$$=(x^2+2y^2)^2-4x^2y^2$$
$$=(x^2+2y^2)^2-(2xy)^2$$
$$=(x^2+2y^2+2xy)(x^2+2y^2-2xy)$$
Now I don't know how this will be helpful because ultimately,  I sill have to compute a large sum for every number. Any body got tips on how I could proceed further to solve this sum? Thanks in advance.

You found:
$$(x^4+4y^4)=(x^2+2y^2+2xy)(x^2+y^2-2xy)$$
Just plug in:
$(x, y)=(10, 3), (4, 3)$
for fraction:
$\frac{10^4+324}{4^4+324}=\frac{178\times 58}{58\times 10}$
$(x, y)=(22, 3), (16, 3)\rightarrow \frac{22^4+324}{16^4+324}=\frac{634\times 370}{370\times178} $
continues this procedure you finally get:
$\frac{178}{10}\times\frac{634}{178}\times\frac{1378}{634}\times \frac{2410}{1378}\times\frac{3730}{2410}$
$\frac {3730}{10}=373$
