(x+1/x)(x^2 + 1/x^2/(x^4 + 1/x^4)(x^8+1/x^8)

==> Let us multiply and divide by x -1/x

==>[ (x-1/x)(x+1/x)(x^2 +1/x^2)(x^4 + 1/x^4)(x^8+ 1/x^8)]/(x1/x)]

=( x^2 -1/x^2)(x^2 + 1/x^2 )(x^4+ 1/x^4)(x^8+1/x^8)]/(x-1/x)

= (x^4 - 1/x^4)(x^4+1/x^4)(x^8+1/x^8)]/(x-1/x)

=(x^8-1/x^8)(x^8+1/x^8)]/(x-1/x)

=(x^16-1/x^16)/(x-1/x)

For simplify the product , we have to multiply it with the factor (x – 1/x)

(x – 1/x)*E(x)=(x – 1/x)* (x + 1/x)* (x ^2+ 1/x^2)* (x ^4+ 1/x^4)* (x ^8+ 1/x^8)

But (x – 1/x)* (x + 1/x)= (x ^2 - 1/x^2)

(x – 1/x)*E(x)= (x ^2 - 1/x^2) * (x ^2+ 1/x^2)* (x ^4+ 1/x^4)* (x ^8+ 1/x^8)

But (x ^2 - 1/x^2) * (x ^2+ 1/x^2)= (x ^4- 1/x^4)

(x – 1/x)*E(x)= (x ^4- 1/x^4) * (x ^4+ 1/x^4)* (x ^8+ 1/x^8)

(x – 1/x)*E(x)= (x ^8- 1/x^8)* (x ^8+ 1/x^8)= (x ^16 - 1/x^16)

**E(x)= (x ^16 - 1/x^16) / (x – 1/x)**

(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)}