Question

cube roots of 64i?

Answer 1

Convert the given number into the form
a+bi = α(cos(β)+i sin(β))
where
α = sqrt(a^2+b^2)
β = sin^-1(b/α)
If the number is plot in the complex (Z) plane, it will be more evident.
For example, 64i will have
α=64 (the distance from origin)
β=90° (sin^-164/64=1)
The equivalent angles are:
90°
450°
810°
(we only need 3)

Divide each of these angles by , to get β1, β2, & β3. And take α'=α^(1/3)=64^(1/3)=4
The three cube roots are then
z1=α'(cos(β1)+isin(β1)
=4(cos(30°)+isin(30°))
z2=4(cos(150°)+isin(150°))
z3=4(cos(270°)+isin(270°))

Check by expanding z1^3, z2^3 and z3^3 to get back 64i.

Answer 2

β=90° (sin-164/64=sin^-11=π/2)