z1 = 3^(1/2) + 3^(1/2) * i
z2 = 2i
z1 + z2 = 3^(1/2) + 3^(1/2) * i + 2i = 3^(1/2) + (3^(1/2) + 2) * i
z1 * z2 = (3^(1/2) + 3^(1/2) * i) * 2i = 2 * 3^(1/2) * i + 2 * 3^(1/2) * i^2 = -2 * 3^(1/2) + 2 * 3^(1/2) * i
z1/z2 = (3^(1/2) + 3^(1/2) * i)/(2i) = ((3^(1/2) + 3^(1/2) * i) * i)/(2i * i) = (3^(1/2) * i + 3^(1/2) * i^2)/(2 * i^2) = (-3^(1/2) + 3^(1/2) * i)/(-2) =
= 3^(1/2)/2 - 3^(1/2)/2 * i
z2^3 = (2i)^3 надеюсь сами уже сможете посчитать
w^3 = z1
w^3 = 3^(1/2) + 3^(1/2) * i
Приведем z1 к тригонометрическому виду.
|z1| = ((3^(1/2))^2 + (3^(1/2))^2)^(1/2) = 6^(1/2)
z1 = 6^(1/2) * (3^(1/2)/6^(1/2) + 3^(1/2)/6^(1/2) * i) = 6^(1/2) * (1/2^(1/2) + 1/2^(1/2) * i) = 6^(1/2) * (cos (pi/4) + sin (pi/4) * i)
w^3 = 6^(1/2) * (cos (pi/4 + 2pi n) + sin (pi/4 + 2pi n) * i)
w = 6^(1/6) * (cos (pi/12 + 2pi n/3) + sin (pi/12 + 2pi n/3) * i)
где n = 0, 1, 2
Получаем, что
w1 = 6^(1/6) * (cos (pi/12) + sin (pi/12) * i)
w2 = 6^(1/6) * (cos (3pi/4) + sin (3pi/4) * i)
w3 = 6^(1/6) * (cos (17pi/12) + sin (17pi/12) * i)