z^4=-8sqrt(3)+8i

解

z^{4} = - 83 + 8 i

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

解答ステップ

z^{4} = - 83 + 8 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 = - 83 + 8 i

∣ a ∣ = 16

ar g (a) = - \frac{π}{6} + π

z = 416 (cos (\frac{- \frac{π}{6} + π + 2 \cdot 0 π}{4}) + i sin (\frac{- \frac{π}{6} + π + 2 \cdot 0 π}{4})), z = 416 (cos (\frac{- \frac{π}{6} + π + 2 \cdot 1 π}{4}) + i sin (\frac{- \frac{π}{6} + π + 2 \cdot 1 π}{4})), z = 416 (cos (\frac{- \frac{π}{6} + π + 2 \cdot 2 π}{4}) + i sin (\frac{- \frac{π}{6} + π + 2 \cdot 2 π}{4})), z = 416 (cos (\frac{- \frac{π}{6} + π + 2 \cdot 3 π}{4}) + i sin (\frac{- \frac{π}{6} + π + 2 \cdot 3 π}{4}))

簡素化

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

簡素化

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

簡素化

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

簡素化

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

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