L.H.S = (1 + i¹⁰ + i²⁰ + i³⁰)

= (1 + + )

since = 1

= i

= -1

= 1 +

= 1 + -1 +1 + -1

= 0, which is a real no.

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