Example. Find all three cube roots of the complex number 110+74i.
(%i1) eq:z^3=polarform(74*%i+110);
(%o1) z^3=26^(3/2)*%e^(%i*atan(37/55))
(%i2) T1:sin(atan(37/55)/3)=1/sqrt(26)$
(%i3) T2:cos(atan(37/55)/3)=5/sqrt(26)$
(%i4) solve(eq,z);
(%o4)
[z=((sqrt(3)*sqrt(26)*%i-sqrt(26))*%e^((%i*atan(37/55))/3))/2,
z=-((sqrt(3)*sqrt(26)*%i+sqrt(26))*%e^((%i*atan(37/55))/3))/2,
z=sqrt(26)*%e^((%i*atan(37/55))/3)]
(%i5) rectform(%)$
(%i6) subst([T1,T2],%),radcan,rectform;
(%o6)
[z=((5*sqrt(3)-1)*%i)/2+(-sqrt(3)-5)/2,z=-((5*sqrt(3)+1)*%i)/2-(5-sqrt(3))/2,z=%i+5]
best
Aleksas D