Find the solutions of the equation. (Give your answers in the form a + bi, where a and b are real numbers. Enter your answers as a comma-separated list.)

Question

Find the solutions of the equation. (Give your answers in the form a + bi, where a and b are real numbers. Enter your answers as a comma-separated list.)

x³ + 343i = 0

Answer 1

Zoe, I would like to tutor you in complex domain and functions if you need help.

I do this problem this way

z=[(-7)³*(cos(π/2+2πk)+i*sin(π/2+2πk))]^(1/3) k=0,1,2 or

z₁=-7(cos(π/6)+i*sin(π/6)) =-7(√3/2+i*1/2)

z₂=-7(cos(5π/6)+i*sin(5π/6) =-7(-√3/2+i*1/2)`

z₃=-7(cos(9π6)+i*sin(9π/6)= -7*-i=7*i

Answer 2

The three cube roots are -7i, -7iω, and -7iω²

where ω = one of the complex cube roots of 1, namely [(-1+3i)/2].

I will leave that complex multiplication to you!