Evaluate:
$$\Large \frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}$$
So I cheated and plugged this in the calculator which gave 373 but I want to know how to evaluate this without a calculator.

Question

lgbasallote · Answer

uhmm you just...

blockcolder · Answer

I thought you had a solution. :|

turingtest · Answer

did you try to write things as multiples of 2 ?
(just brainstorming...)

lgbasallote · Answer

seems like a case of $$\large \frac{[n^4 + 324][(n+12)^4 + 324][(n+24)^4 + 324]...[(n+48)^4 + 324]}{[m^4 +324)][(m+12)^4 + 324][(m+24)^24 + 324]...[(m+48)^4 + 324]}$$

turingtest · Answer

but doesn't it seems suspicious that all the bases are even?
I wanna try to write 10^4=(5*2)^4 and factor out that 2^4 ....

lgbasallote · Answer

this seems like one of them thingies with the |dw:1337259106151:dw|

turingtest · Answer

whatever, I'm terrible at these kinds of problems anyway :P

anonymous · Answer

This seems like getting out a TI-84 and abusing the parentheses button... 

Just type it all in....

anonymous · Answer

Oh, i just read your note about the calculator...

kinggeorge · Answer

My first instinct is to somehow use the fact that $324=18^2$

anonymous · Answer

I believe this is how you would do it by hand.

anonymous · Answer

two things, the nice factorization, and persistence.  Had no idea it was going to work until that division  near  the end.

blockcolder · Answer

So in the end, it was just a telescoping product. Thanks a lot!