Question

Las raíces cubicas de 8i

Answer 1

La raíz enésima de un complejo es:

z^(1/n) = |z|^(1/n) {cos[(Ф + 2kπ)/n + i sen[(Ф 2kπ)/n]}

|z| = 8; 8^(1/3) = 2; n = 3

Ф = π/2, k = 0, 1, 2

k = 0; zo = 2 {cos[(π/2)/3] + i sen[(π/2)/3]} = 2 {√3/2 + i 1/2] = √3 + i

k = 1; z1 = 2 {cos[(π/2 + 2π)/3] + i sen[(π/2 + 2π)/3] = - √3 + i

k = 2; z2 = 2 {cos[(π/2 + 4π)/3] + i sen[(π/2 + 4π)/3]} = - 2 i

Saludos Herminio