Question

Evaluate (-1+i(root 3))^60

Answer 1

2^60.

Explanation:

You can do this using deMoivres theorem.

First convert to Polar form:

(-1+i(root 3))^60

r = √((-1)^2 + (√3)^2)

= √(1 + 3)

= 2.

So sin Ф = √3/2

Ф = 2π/3 - as the angle is in the second quadrant ( real part is -1)

So in polar form we have:

[2(cos2π/3 + i sin2π/3 )]^60

Bu de Moivres theorem this is

2^60(cos2π/3* 60 + i sin2π/3* 60)

= 2^60(cos(120π/3) + i sin(120π/3)

= 2^60(cos40π + i sin 40π)

= 2^60(1 + i*0)

= 2^60.