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 = 2 cos (\frac{π}{30}) + 2 i sin (\frac{π}{30}), z = 2 cos (\frac{13 π}{30}) + 2 i sin (\frac{13 π}{30}), z = - 3 + i, z = 2 cos (\frac{37 π}{30}) + 2 i sin (\frac{37 π}{30}), z = 2 cos (\frac{49 π}{30}) + 2 i sin (\frac{49 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}

Simplify

z = 2 cos (\frac{π}{30}) + 2 i sin (\frac{π}{30}), z = 2 cos (\frac{13 π}{30}) + 2 i sin (\frac{13 π}{30}), z = - 3 + i, z = 2 cos (\frac{37 π}{30}) + 2 i sin (\frac{37 π}{30}), z = 2 cos (\frac{49 π}{30}) + 2 i sin (\frac{49 π}{30})

15 = k^{3}

12 y^{2} - 13 y^{3} = 2 y^{2} - 2 y

x^{813} = 2

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

16 x^{3} - x = 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