Let A a be the angle so that the polar form of z1+z2 is (20,A).
Let B a be the angle so that the polar form of z1^2+z2^2 is (16,B).
Create an identity to express z1^3+z2^3 in terms of z1+z2 and z1^2+z2^2:
z1^3+z2^3 = (z1+z2)*(1.5*(z1^2+z2^2)-0.5(z1+z2)^2)
Then substitute and do some complex arithemetic:
z1^3+z2^3 = (20,A)*(1.5*(16,B)-0.5*(20,A)^2)
= (20,A)*((24,B)-(400,2A))
= (20,A)*(1,B)*(24-(200,2A-B))
Then reintroduce the absolute values:
|z1^3+z2^3| = |(20,A)*(1,B)*(24-(200,2A-B))|
= |(20,A)|*|(1,B)|*|(24-(200,2A-B))|
= 20*|24-(200,2A-B)|
Taking 2A-B=pi will maximize |24-(200,2A-B)|. Then the maximum value of |z1^3+z2^3| = 20*(24 - (-200)) = 4480.
Solution
