Question 1074318
<pre>
{{{-64i = 64(cos("270°"+"360*"n)^""+i*sin("270°"+"360*"n))}}}

Therefore:

{{{matrix(2,3,"","","",
(-64i)^(1/3),""="",64^(1/3)(cos("270°"+"360*"n)^""+i*sin("270°"+"360*"n))^(1/3))}}}

We use DeMoivre's theorem to multiply the angle by the exponent {{{1/3}}}:

{{{matrix(2,3,"","","",
(-64i)^(1/3),""="",root(3,64)(cos(expr(1/3)("270°"+"360*"n))^""+i*sin(expr(1/3)("270°"+"360*"n))))}}}


{{{matrix(2,3,"","","",
(-64i)^(1/3),""="",4(cos("90°"+"120°"n)^""+i*sin("90°"+"120°"n)))}}}

We can use any three consecutive integers for n to obtain the
three cube roots:

Using n=0

{{{matrix(2,7,"","","","","","","",
4(cos("90°"+"120°"0)^""+i*sin("90°"+"120°"0)),""="",4(cos("90°")^""+i*sin("90°")),""="",4(0+i*1),""="",4i  )}}}

Using n=1

{{{matrix(2,7,"","","","","","","",
4(cos("90°"+"120°"*1)^""+i*sin("90°"+"120°"*1)),""="",4(cos("210°")^""+i*sin("210°")),""="",4(-sqrt(3)/2+i*(expr(-1/2))^""),""="",-2sqrt(3)-2i  )}}}

Using n=2

{{{matrix(2,7,"","","","","","","",
4(cos("90°"+"120°"*2)^""+i*sin("90°"+"240°"*1)),""="",4(cos("330°")^""+i*sin("330°")),""="",4(sqrt(3)/2+i*(expr(-1/2))^""),""="",2sqrt(3)-2i  )}}}

So the three cube roots of -64i are

{{{4i}}}, {{{-2sqrt(3)-2i}}}, and {{{2sqrt(3)-2i}}}

Edwin</pre>