ω(13 + 13i) = 2 + √13 i
ω^2(13 + 13i) = -1 + √13 i
ω^3(13 + 13i) = 13 - 13i
ω(13 + 13i) = 2 + √13 i, where ω is a complex cube root of 1.
ω^2(13 + 13i) = -1 + √13 i.
ω^3(13 + 13i) = 13 - 13i.
These can be found using DeMoivre's Theorem, which states that for any complex number z and any positive integer n:
(cis θ)^n = cis nθ
where cis θ is the complex number in polar form with angle θ.
In this case, z = 13 + 13i and n = 3. Therefore, the cube roots of z are:
ωz = cis(θ/3), ω^2z = cis(2θ/3), and ω^3z = cisθ.
The angle θ can be found by solving the equation z = cis θ. In this case, z = 13 + 13i has angle θ = arctangent(13/13) = π/4.
Therefore, the cube roots of z are:
ωz = cis(π/12), ω^2z = cis(π/2), and ω^3z = cis(π/4).
These can then be converted to rectangular form using the following formulas:
cis θ = cos θ + i sin θ
sin θ = 2 / (1 + e^(-2iθ))
cos θ = (1 + e^(-2iθ)) / 2