1 + i¹⁰ + i²⁰ + i³⁰ = 1 + i^{(8 + 2) +} i²⁰ + i^{(28 + 2)}

= 1 + (i⁴)² × i² + (i⁴)⁵ + (i⁴)⁷ × i²

= 1 – 1 + 1 – 1 = 0

[ since i⁴ = 1, i² = – 1]

Hence , 1 + i¹⁰ + i²⁰ + i³⁰ is a real number.