Find all three cubic roots of −64i. Write all complex numbers in the form a + b i; use exact values.

Answer 1

In polar form, -64i = 64(cos270° + isin270°)

Using DeMoivre's Theorem, one of the cube roots is 64^1/3[cos(270°/3 + isin(270°/3)] = 4[cos90° + isin90°] = 4i

The 3 cube roots of -64i are equally spaced around the circle with center (0,0) and radius 4.

360°/3 = 120°

The other two cube roots are:

4[cos(90°+120°) + isin(90°+120°)] = 4[cos210° + isin210°] = 4[-√3/2 - (1/2)i] = -2√3 - 2i

and

4[cos(210°+120°) + isin(210°+120°)] = 4[cos330° + isin330°] = 4[√3/2 - (1/2)i] = 2√3 - 2i

Answer 2

The cube root of 64 is 4

That leaves you with the cube root of -i. For this, use this table:

n iⁿ

0 +1

1 +i

2 -1

3 -i

4 +1

5 +i

6 -1

etc. (Notice the pattern)