What are the solutions to the equation z^5=sqrt(2)-sqrt(2)i ?

The solutions to the equation z^5=sqrt(2)-sqrt(2)i are

Find the zeros of z^5=sqrt(2)-sqrt(2)i

The zeros of z^5=sqrt(2)-sqrt(2)i are

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z^{5} = 2 - 2 i

z = \frac{2 ^{\frac{7}{10}} 4 + 10 + 2 5}{4} - i \frac{2 ^{\frac{7}{10}} 4 - 10 + 2 5}{4}, z = \frac{2 ^{\frac{7}{10}} 4 - 10 - 2 5}{4} + i \frac{2 ^{\frac{7}{10}} 4 + 10 - 2 5}{4}, z = - \frac{2 ^{\frac{7}{10}}}{2} + i \frac{2 ^{\frac{7}{10}}}{2}, z = - \frac{2 ^{\frac{7}{10}} 4 + 10 - 2 5}{4} - i \frac{2 ^{\frac{7}{10}} 4 - 10 - 2 5}{4}, z = \frac{2 ^{\frac{7}{10}} 4 - 10 + 2 5}{4} - i \frac{2 ^{\frac{7}{10}} 4 + 10 + 2 5}{4}

Solution steps

z^{5} = 2 - 2 i

For

z^{n} = a

the solutions are

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

For

n = 5, a = 2 - 2 i

∣ a ∣ = 2

ar g (a) = - \frac{π}{4}

Simplify

z = \frac{2 ^{\frac{7}{10}} 4 + 10 + 2 5}{4} - i \frac{2 ^{\frac{7}{10}} 4 - 10 + 2 5}{4}, z = \frac{2 ^{\frac{7}{10}} 4 - 10 - 2 5}{4} + i \frac{2 ^{\frac{7}{10}} 4 + 10 - 2 5}{4}, z = - \frac{2 ^{\frac{7}{10}}}{2} + i \frac{2 ^{\frac{7}{10}}}{2}, z = - \frac{2 ^{\frac{7}{10}} 4 + 10 - 2 5}{4} - i \frac{2 ^{\frac{7}{10}} 4 - 10 - 2 5}{4}, z = \frac{2 ^{\frac{7}{10}} 4 - 10 + 2 5}{4} - i \frac{2 ^{\frac{7}{10}} 4 + 10 + 2 5}{4}

Z^{3} + 4 = 5 + i

x^{6} - 10 x^{3} + 24 = 0

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

x^{3} - 3 x^{2} + 3 x - 65 = 0

(x^{2} + 6)^{2} = 0

What are the solutions to the equation z^5=sqrt(2)-sqrt(2)i ?
The solutions to the equation z^5=sqrt(2)-sqrt(2)i are
Find the zeros of z^5=sqrt(2)-sqrt(2)i
The zeros of z^5=sqrt(2)-sqrt(2)i are