Question

Which expression is equivalent to (sqrt3 + i)^3?

Answer 1

Express √3 + i in polar form:

|√3 + i| = √((√3)² + 1²) = √4 = 2

arg(√3 + i) = arctan(1/√3) = π/6

Then

√3 + i = 2 (cos(π/6) + i sin(π/6))

By DeMoivre's theorem,

(√3 + i)³ = (2 (cos(π/6) + i sin(π/6)))³

… = 2³ (cos(3 • π/6) + i sin(3 • π/6))

… = 8 (cos(π/2) + i sin(π/2))

… = 8i