AI Hallucination observed: wrong answer was given by Copilot agent on its first try. The error was effectively corrected by applying Prompt Engineering technique during the interactive AI Copilot session.
𝑥=±(1±𝑖)/√2 (the answer).
Better alternative to Copilot's workings is Polynomial factoring:
𝑥⁴+1 = (𝑥²+1)²−2𝑥² = (𝑥²+𝑥√2+1)(𝑥²-𝑥√2+1) = 0
Solving 2 quadratic equations (𝑥²+𝑥√2+1)=0 and (𝑥²-𝑥√2+1)=0 gave 4 complex roots written in a compact Math notation 𝑥=±(1±𝑖)/√2.
k¹⁵=-1 is the answer to this Math problem.
Having (k²-k+1)=0; (k³+1)=(k+1)(k²-k+1)=0, thence:
k³+1=0, k³=-1 and k¹⁵=(k³)⁵=-1
Btw, same goes to: k²⁰²⁵=-1