What are the solutions to the equation z^5=-16sqrt(3)+16i ?

The solutions to the equation z^5=-16sqrt(3)+16i are

Find the zeros of z^5=-16sqrt(3)+16i

The zeros of z^5=-16sqrt(3)+16i are

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z^{5} = - 163 + 16 i

z = 3 + i, z = 2 cos (\frac{17 π}{30}) + 2 i sin (\frac{17 π}{30}), z = 2 cos (\frac{29 π}{30}) + 2 i sin (\frac{29 π}{30}), z = 2 cos (\frac{41 π}{30}) + 2 i sin (\frac{41 π}{30}), z = 2 cos (\frac{53 π}{30}) + 2 i sin (\frac{53 π}{30})

Solution steps

z^{5} = - 163 + 16 i

For

z^{n} = a

the solutions are

k = 0, 1, \dots, n - 1

For

n = 5, a = - 163 + 16 i

∣ a ∣ = 32

ar g (a) = - \frac{π}{6} + π

Simplify

z = 3 + i, z = 2 cos (\frac{17 π}{30}) + 2 i sin (\frac{17 π}{30}), z = 2 cos (\frac{29 π}{30}) + 2 i sin (\frac{29 π}{30}), z = 2 cos (\frac{41 π}{30}) + 2 i sin (\frac{41 π}{30}), z = 2 cos (\frac{53 π}{30}) + 2 i sin (\frac{53 π}{30})

x^{3} = 343

8 x^{3} - 27 = 0

x^{3} + 3 x^{2} + 2 x + 4 = 0

4 x^{3} - 12 x^{2} = 0

x^{3} - 216 = 0

What are the solutions to the equation z^5=-16sqrt(3)+16i ?
The solutions to the equation z^5=-16sqrt(3)+16i are
Find the zeros of z^5=-16sqrt(3)+16i
The zeros of z^5=-16sqrt(3)+16i are