Question 1104069
Here is a representation of {{{4sqrt(2)+i4sqrt(2)}}}
{{{drawing(300,300,-10,10,-10,10,grid(0),
triangle(0,0,5.657,5.657,5.657,0),
arc(0,0,6,6,-45,0),locate(1.8,1.5,theta),
locate(3.5,3.5,r),circle(5.657,5.657,0.2),
locate(5.7,7,4sqrt(2)+i4sqrt(2))
)}}} {{{4sqrt(2)+i4sqrt(2)=r(cos(theta)+i*sin(theta))}}} 
Obviously, {{{tan(theta)=4sqrt(2)/4sqrt(2)=1}}} , so {{{theta=pi/4}}} .
In degrees, {{{theta=45^o}}} .
{{{r=sqrt((4sqrt(2))^2+(4sqrt(2))^2)=sqrt(32+32)=sqrt(64)=8}}}
So,
{{{4sqrt(2)+i4sqrt(2)=8(cos(pi/4)+i*sin(pi/4))}}} .
There are {{{3}}} complex values for {{{(8(cos(pi/4)+i*sin(pi/4)))^(1/3)}}} .
They are complex numbers of the form {{{r(cos(theta)+i*sin(theta))}}}
such that {{{(matrix(3,1," ",r(cos(theta)+i*sin(theta))," "))^3=8(cos(pi/4)+i*sin(pi/4))}}} .
So, it will be {{{r^3=8}}} and {{{3theta=pi/4+k(2pi)}}} for {{{"k = 0 , 1 , 2"}}} .
In degrees, {{{3theta=45^o+k360^o}}} .
That means {{{r=2}}} and
{{{system(theta=pi/12,theta=3pi/4,theta=17pi/12)}}} .
In degrees, {{{system(theta=45^o/3=15^o,theta=(45^o+360^o)/3=405^o/3=135^o,theta=(45^o+720^o)/3=765^o/3=255^o)}}} .
The answers could be written as 
{{{green(Z[1])=2cos(15^o)+i*sin(15^o)}}} ,{{{red(Z[2])=2cos(135^o)+i*sin(135^o)}}} , and {{{blue(Z[3])=2cos(255^o)+i*sin(255^o)}}} 
(or you could express the angles in radians).
They can be represented as
{{{drawing(900,900,-4,7,-4,7,grid(0),
triangle(0,0,5.657,5.657,5.657,0),
arc(0,0,6,6,-45,0),locate(2.5,2.2,pi/4),
locate(3.5,3.5,8),circle(5.657,5.657,0.1),
locate(5.7,6.2,4sqrt(2)+i4sqrt(2)),
green(triangle(0,0,1.932,0.518,1.932,0)),
arrow(1.932,0.518,6.761,1.812),locate(1.5,0.4,green(2)),
locate(1.8,1,green(Z[1])),green(circle(1.932,0.518,0.1)),
green(arc(0,0,8,8,-15,0)),locate(4,1,green(pi/12)),
arrow(0,0,-2.828,2.828),arrow(0,0,-1.035,-3.864),
red(triangle(0,0,-1.414,1.414,-1.414,0)),
locate(-1,0.9,red(2)),red(arc(0,0,2,2,-135,0)),
locate(-0.4,0.9,red(3pi/4)),blue(arc(0,0,7,7,105,360)),
locate(-0.35,-0.7,blue(2)),locate(-3,-1.5,blue(17pi/12)),
red(circle(-1.414,1.414,0.1)),locate(-1.5,1.9,red(Z[2])),
blue(triangle(-0.518,-1.932,-0.518,0,0,0)),
blue(circle(-0.518,-1.932,0.1)),locate(-0.8,-1.8,blue(Z[3]))
)}}}