If (x ) is a cube root of unity other than 1 , then ( (x+ 1 x )^2+ (x^2+ 1 x^2 )^2+ + (x¹²+ 1 x¹² )^2= )

Step-by-step solution

Given, (x ) is a cube root of unity other than 1 i.e. (x= ) or ( ^2 ) Now ( aligned & ( (x+ 1 x )^2+ (x^2+ 1 x^2 )^2+ + (x¹²+ 1 x¹² )^2 . & = ( + 1 )^2+ ( ^2+ 1 ^2 )^2+ + ( ¹²+ 1 ¹² )^2 & = ( + 1 )^2+ ( ^2+ 1 ^2 )^2+ + ( ¹¹+ 1 ¹¹ )^2 & = ( + ^2 )^2+ ( ^2+ )^2+(1+1)^2+ ( + ^2 )^2 & + ( ^2+ )^2+(1+1)^2+ ( + ^2 )^2 & =8 ( + ^2 )^2+4(1+1)^2 & =8(-1)^2+4 (2^2=8+16=24 . aligned )