Finding The Cube Root(s) of 18 + 26i | Problem 287

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[@SweetSorrow777]

Writing the problem in polar form could produce those other solutions.

[@ibnSafaa]

you can easliy use the cubic roots of unity (w , w² , 1)
x³=a³
∴x=a , x=aw , x=aw²

[@DonRedmond-jk6hj]

How does your cube root formula compare with DeMoivre's formula?

[@krabbediem]

Loved it! Thank you :D

[@mikelivstone]

Do the second and third solutions correspond to the 2 remaining values of t at 3:53? That avenue was not really pursued.

[@Nobodyman181]

a,b,t,k
Too many constant 😅

[@bkkboy-cm3eb]

α=³√(18+26i), β=³√(18-26i)
α³+β³=36
α³β³=1000 → αβ=10
(α+β)³=α³+β³+3αβ(α+β)
α+β=t → t³-30t-36=0
(t-6)(t²+6t+6)=0 → t=6
∴α+β=6, αβ=10 → α²-6α+10=0
∴α=3+i
α₁=α·e^(i2π/3)
α₂=α·e^(i4π/3)
∴α=3+i, (-3-√3+(3√3-1)i)/2, (3+√3+(1-3√3)i)/2

[@scottleung9587]

Cool!

[@phill3986]

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