What are the solutions to the equation z^4=16sqrt(3)-16i ?

The solutions to the equation z^4=16sqrt(3)-16i are

Find the zeros of z^4=16sqrt(3)-16i

The zeros of z^4=16sqrt(3)-16i are

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z^{4} = 163 - 16 i

z = \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{2} - i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{2}, z = \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{2} + i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{2}, z = - \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{2} + i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{2}, z = - \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{2} - i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{2}

Solution steps

z^{4} = 163 - 16 i

For

z^{n} = a

the solutions are

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

For

n = 4, a = 163 - 16 i

∣ a ∣ = 32

ar g (a) = - \frac{π}{6}

Simplify

z = \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{2} - i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{2}, z = \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{2} + i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{2}, z = - \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{2} + i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{2}, z = - \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{2} - i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{2}

216 z^{3} - 1 = 0

3 x^{3} + 5 x^{2} = 2 x

2 x^{3} - x^{2} - 21 x = 0

x^{4} - 20 x^{2} + 100 = 0

x^{3} + 12 x^{2} = - 35 x

What are the solutions to the equation z^4=16sqrt(3)-16i ?
The solutions to the equation z^4=16sqrt(3)-16i are
Find the zeros of z^4=16sqrt(3)-16i
The zeros of z^4=16sqrt(3)-16i are