What are the solutions to the equation z^4=4-i ?

The solutions to the equation z^4=4-i are

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z^4=4-i

z^{4} = 4 - i

z = 817 cos (\frac{arctan ( \frac{1}{4} )}{4}) - 817 i sin (\frac{arctan ( \frac{1}{4} )}{4}), z = 817 cos (\frac{- arctan ( \frac{1}{4} ) + 2 π}{4}) + 817 i sin (\frac{- arctan ( \frac{1}{4} ) + 2 π}{4}), z = 817 cos (\frac{- arctan ( \frac{1}{4} ) + 4 π}{4}) + 817 i sin (\frac{- arctan ( \frac{1}{4} ) + 4 π}{4}), z = 817 cos (\frac{- arctan ( \frac{1}{4} ) + 6 π}{4}) + 817 i sin (\frac{- arctan ( \frac{1}{4} ) + 6 π}{4})

Solution steps

z^{4} = 4 - i

For

z^{n} = a

the solutions are

z_{k} = n ∣ a ∣ (cos (\frac{ar g ( a ) + 2 kπ}{n}) + i sin (\frac{ar g ( a ) + 2 kπ}{n})),

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

For

n = 4, a = 4 - i

∣ a ∣ = 17

ar g (a) = - arctan (\frac{1}{4})

z = 417 (cos (\frac{- arctan ( \frac{1}{4} ) + 2 \cdot 0 π}{4}) + i sin (\frac{- arctan ( \frac{1}{4} ) + 2 \cdot 0 π}{4})), z = 417 (cos (\frac{- arctan ( \frac{1}{4} ) + 2 \cdot 1 π}{4}) + i sin (\frac{- arctan ( \frac{1}{4} ) + 2 \cdot 1 π}{4})), z = 417 (cos (\frac{- arctan ( \frac{1}{4} ) + 2 \cdot 2 π}{4}) + i sin (\frac{- arctan ( \frac{1}{4} ) + 2 \cdot 2 π}{4})), z = 417 (cos (\frac{- arctan ( \frac{1}{4} ) + 2 \cdot 3 π}{4}) + i sin (\frac{- arctan ( \frac{1}{4} ) + 2 \cdot 3 π}{4}))

Simplify

417 (cos (\frac{- arctan ( \frac{1}{4} ) + 2 \cdot 0 π}{4}) + i sin (\frac{- arctan ( \frac{1}{4} ) + 2 \cdot 0 π}{4})) : 817 cos (\frac{arctan ( \frac{1}{4} )}{4}) - 817 i sin (\frac{arctan ( \frac{1}{4} )}{4})

Simplify

417 (cos (\frac{- arctan ( \frac{1}{4} ) + 2 \cdot 1 π}{4}) + i sin (\frac{- arctan ( \frac{1}{4} ) + 2 \cdot 1 π}{4})) : 817 cos (\frac{- arctan ( \frac{1}{4} ) + 2 π}{4}) + 817 i sin (\frac{- arctan ( \frac{1}{4} ) + 2 π}{4})

Simplify

417 (cos (\frac{- arctan ( \frac{1}{4} ) + 2 \cdot 2 π}{4}) + i sin (\frac{- arctan ( \frac{1}{4} ) + 2 \cdot 2 π}{4})) : 817 cos (\frac{- arctan ( \frac{1}{4} ) + 4 π}{4}) + 817 i sin (\frac{- arctan ( \frac{1}{4} ) + 4 π}{4})

Simplify

417 (cos (\frac{- arctan ( \frac{1}{4} ) + 2 \cdot 3 π}{4}) + i sin (\frac{- arctan ( \frac{1}{4} ) + 2 \cdot 3 π}{4})) : 817 cos (\frac{- arctan ( \frac{1}{4} ) + 6 π}{4}) + 817 i sin (\frac{- arctan ( \frac{1}{4} ) + 6 π}{4})

z = 817 cos (\frac{arctan ( \frac{1}{4} )}{4}) - 817 i sin (\frac{arctan ( \frac{1}{4} )}{4}), z = 817 cos (\frac{- arctan ( \frac{1}{4} ) + 2 π}{4}) + 817 i sin (\frac{- arctan ( \frac{1}{4} ) + 2 π}{4}), z = 817 cos (\frac{- arctan ( \frac{1}{4} ) + 4 π}{4}) + 817 i sin (\frac{- arctan ( \frac{1}{4} ) + 4 π}{4}), z = 817 cos (\frac{- arctan ( \frac{1}{4} ) + 6 π}{4}) + 817 i sin (\frac{- arctan ( \frac{1}{4} ) + 6 π}{4})

2 x^{4} - 5 x^{3} - 20 x^{2} + 115 x - 52 = 0

4 x^{3} - 36 x^{2} + 104 x - 96 = 0

z^{3} = (- 3 + i)

2 x^{3} + 2 x^{2} - 19 x + 20 = 0

3 x^{3} + 7 x^{2} - 4 = 0

What are the solutions to the equation z^4=4-i ?
The solutions to the equation z^4=4-i are
Find the zeros of z^4=4-i
The zeros of z^4=4-i are