Step-by-step explanation:
one of the cube roots (there are 3 of them) is
81/3[cos(270°/3) + isin(270°/3)] = 2(cos90° + isin90°) = 2i
The 3 cube roots are equally spaced around the circle centered at (0,0) with radius 81/3 = 2
360° / 3 = 120°
Another cube root of -8i is
2[cos(90° + 120°) + isin(90° + 120°)] = 2[cos(210°) + isin(210°)] = 2(-√3/2 + i(-1/2)) = -√3 - i
The third cube root of -8i is
2[cos(90°+2(120°)) + isin(90°+2(120°)) = 2[cos330° + isin(330°)] = 2[√3/2 +i(-1/2)] = √3 - i
So, the cube roots of -8i are 2i, -√3 - i, and √3 - i
