The four roots of the complex number are √2cis(-π/24), √2cis(11π/24), √2cis(23π/24), and √2cis(35π/24) in increasing order of the angles.
The complex number given to us is 2√3 - 2i. We have to find the fourth roots of the given complex number. First, we will convert the given complex number into polar form.
Z = 2√3 - 2i
Z = R(cisθ)
R = |Z|
R = √[(2√3)² + (-2)²]
R = √(12 + 4)
R = 4
Tanθ = -2/2√3
Tanθ = -1/√3
θ = -30°
Z = 4(cis(-π/6 + 2kπ))
To find the fourth roots. Let s be the roots.
z^4 = Z
r^4[cis4α] = 4(cis(-π/6 + 2kπ))
r^4 = 4
r = √2
4α = -π/6 + 2kπ
α = -π/24 + kπ/2
As the values of k vary from 0 to 3, we get the four roots.
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