Find all complex number solutions for x^3+i=0.

Question

Find all complex number solutions for x^3+i=0.

Answer 1

De Moivre's Theorem

 

zⁿ = rⁿ (cos nθ + i sin nθ), where z = complex number, r = modulus, n = power, θ = angle

 

x³ + i = 0

x³ = -i

x³ = (0 - 1i)

x = (0 - i)^1/3

 

r = 1

θ = 3π/2

n = 1/3

 

(-i)^1/3 = 1^1/3 { cos [(1/3)(3π/2)] + i sin [(1/3)(3π/2)] }

(-i)^1/3 = cos π/2 + i sin π/2

(-i)^1/3 = 0 + i

(-i)^1/3 = i

 

Check:

 

i · i · i

-1 i

-i

Answer 2

Robert, the easy way to do this problem is to convert the imaginary number to its Euler equivalent.  

 

-i=e^(i*pi/2) Hence,

 

x^3+i=0

 

x^3=-i=e^(-i*pi/2) Raise both sides to the 1/3 rd power. 

 

x=(e^(-i*pi/2)^1/3=e^(-i*pi/6)=cos(pi/6)-i*sin(pi/6)

 

x=1/2*√3-i*1/2