Question 1209571
<font color=black size=3>
Answer: <font color=red>2</font>


Explanation


We can pair up these two terms to get:
1/(1+x) + 1/(1+x^4) 
= (1+x^4)/( (1+x)(1+x^4) )  + (1+x)/( (1+x)(1+x^4) )
= (1+x^4+1+x)/(1+x^4+x+x^5)
= (x^4+x+2)/(1+x^4+x+1) ..... use x^5 = 1
= (x^4+x+2)/(x^4+x+2)
= 1


Through similar steps,
1/(1+x^2) + 1/(1+x^3) = (x^3+x^2+2)/(x^5+x^3+x^2+1)
Plug in x^5 = 1 then you'll have it simplify to,
(x^3+x^2+2)/(x^5+x^3+x^2+1)
= (x^3+x^2+2)/(1+x^3+x^2+1)
= (x^3+x^2+2)/(x^3+x^2+2)
= 1


We determined that
1/(1+x) + 1/(1+x^4) = 1 when x^5 = 1
1/(1+x^2) + 1/(1+x^3) = 1 when x^5 = 1



Adding those results gets to the final answer of 2.


--------------------------------------------------------------------------


Another approach:


x^5 = 1 can be solved using the roots of unity formula
See this link for more info
https://artofproblemsolving.com/wiki/index.php/Roots_of_unity
One of the five roots is x = 1, but we cannot use it since the instructions state as such.


But you can use any of the four complex roots of the form a+bi.
Let's say those four roots are p,q,r,s
If you select a complex root at random, say p, then you can compute 1/(1+p^2)+1/(1+p^4)+1/(1+p)+1/(1+p^3) by using a calculator.
The result my calculator says is roughly 2+9.9836*10^(-13)*i
The 9.9836*10^(-13) portion is so very small that it is basically zero. This is due to rounding error. 
So we really have 2+0i or simply 2 as the final answer.
</font>