Simplify the product (x+1/x)(x^2+1/x^2)(x^4+1/x^4)(x^8+1/x^8)

(x+1/x)(x^2+1/x^2) =
x^3+x+1/x+1/x^3

(x4+1/x^4)(x^8+1/x^8) =
x^12+x^+x^4+1/x^4+1/x^12

Thereore,

(x^3+x+1/x+1/x^3)
((x^12+x^4+1/x^4+1/x^12)

= x^15 + x^7 + 1/x +
1/x^9

+x^13 + x^5 + 1/x^3 +
1/x^11

+x^11 + x^3 + 1/x5
+1/x^13

+x^9 + x^1 + 1/x^7+ 1/x^15 ....
(1)

= sum  of  the geometric  progression x^(-15)
+x^(-13)+x^(-11)+x^(-9)+....+x^13 +x^15 m whose artin tern a  = x^-15 and common ratio
is r = x^2 and last term = x^15 and number od terms n =
16

Therefore sum of the G.P at (1) = a (1-r^n)/(1-r) =
x^(-15) { 1- (x^2)^16}/ (1-x^2)

= (1-x^32)/{x^15(1-x^2)} =
(x^32 - 1)/{x^15(x^1-1)}