Let and be the roots of the equation z^2 - kz - 2 i = 0 , where k is a real number and i = -1 . If Re ( ¹⁴ + ¹⁴ + 16 ^6 + 16 ^6 ¹⁰ + ¹⁰ ) = 73 , then the value of k^4 is equal to

Step-by-step solution

Given the equation z^2 - kz - 2 i = 0 , we can rearrange it as: z^2 - 2 i = kz Dividing by z , we get: z - 2 i z = k Squaring both sides: (z - 2 i z )^2 = k^2 z^2 + 4 i ^2 z^2 - 2(z) ( 2 i z ) = k^2 z^2 - 4 z^2 - 4 i = k^2 z^2 - 4 z^2 = k^2 + 4 i Squaring both sides again: (z^2 - 4 z^2 )^2 = (k^2 + 4 i )^2 z^4 + 16 z^4 - 2(z^2) ( 4 z^2 ) = k^4 + 16 i ^2 + 8 i k^2 z^4 + 16 z^4 - 8 = k^4 - 16 + 8 i k^2 z^4 + 16 z^4 = k^4 - 8 + 8 i k^2 Since and are roots of the equation, they both satisfy this relation. Now, consider