z^4=16sqrt(3)-16i

解

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}

解答ステップ

z^{4} = 163 - 16 i

z^{n} = a

の場合, 解は

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

以下のため:

n = 4, a = 163 - 16 i

∣ a ∣ = 32

ar g (a) = - \frac{π}{6}

z = 432 (cos (\frac{- \frac{π}{6} + 2 \cdot 0 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 0 π}{4})), z = 432 (cos (\frac{- \frac{π}{6} + 2 \cdot 1 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 1 π}{4})), z = 432 (cos (\frac{- \frac{π}{6} + 2 \cdot 2 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 2 π}{4})), z = 432 (cos (\frac{- \frac{π}{6} + 2 \cdot 3 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 3 π}{4}))

簡素化

432 (cos (\frac{- \frac{π}{6} + 2 \cdot 0 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 0 π}{4})) : \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{2} - i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{2}

簡素化

432 (cos (\frac{- \frac{π}{6} + 2 \cdot 1 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 1 π}{4})) : \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{2} + i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{2}

簡素化

432 (cos (\frac{- \frac{π}{6} + 2 \cdot 2 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 2 π}{4})) : - \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{2} + i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{2}

簡素化

432 (cos (\frac{- \frac{π}{6} + 2 \cdot 3 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 3 π}{4})) : - \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}, z = - \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{2} - i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{2}