z^4=2+2i

解

z^{4} = 2 + 2 i

z = \frac{2 ^{\frac{3}{8}} 2 + 2 + 2}{2} + i \frac{2 ^{\frac{3}{8}} 2 - 2 + 2}{2}, z = - \frac{2 ^{\frac{3}{8}} 2 - 2 + 2}{2} + i \frac{2 ^{\frac{3}{8}} 2 + 2 + 2}{2}, z = - \frac{2 ^{\frac{3}{8}} 2 + 2 + 2}{2} - i \frac{2 ^{\frac{3}{8}} 2 - 2 + 2}{2}, z = \frac{2 ^{\frac{3}{8}} 2 - 2 + 2}{2} - i \frac{2 ^{\frac{3}{8}} 2 + 2 + 2}{2}

解答ステップ

z^{4} = 2 + 2 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 = 2 + 2 i

∣ a ∣ = 22

ar g (a) = \frac{π}{4}

z = 422 (cos (\frac{\frac{π}{4} + 2 \cdot 0 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 0 π}{4})), z = 422 (cos (\frac{\frac{π}{4} + 2 \cdot 1 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 1 π}{4})), z = 422 (cos (\frac{\frac{π}{4} + 2 \cdot 2 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 2 π}{4})), z = 422 (cos (\frac{\frac{π}{4} + 2 \cdot 3 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 3 π}{4}))

簡素化

422 (cos (\frac{\frac{π}{4} + 2 \cdot 0 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 0 π}{4})) : \frac{2 ^{\frac{3}{8}} 2 + 2 + 2}{2} + i \frac{2 ^{\frac{3}{8}} 2 - 2 + 2}{2}

簡素化

422 (cos (\frac{\frac{π}{4} + 2 \cdot 1 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 1 π}{4})) : - \frac{2 ^{\frac{3}{8}} 2 - 2 + 2}{2} + i \frac{2 ^{\frac{3}{8}} 2 + 2 + 2}{2}

簡素化

422 (cos (\frac{\frac{π}{4} + 2 \cdot 2 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 2 π}{4})) : - \frac{2 ^{\frac{3}{8}} 2 + 2 + 2}{2} - i \frac{2 ^{\frac{3}{8}} 2 - 2 + 2}{2}

簡素化

422 (cos (\frac{\frac{π}{4} + 2 \cdot 3 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 3 π}{4})) : \frac{2 ^{\frac{3}{8}} 2 - 2 + 2}{2} - i \frac{2 ^{\frac{3}{8}} 2 + 2 + 2}{2}

z = \frac{2 ^{\frac{3}{8}} 2 + 2 + 2}{2} + i \frac{2 ^{\frac{3}{8}} 2 - 2 + 2}{2}, z = - \frac{2 ^{\frac{3}{8}} 2 - 2 + 2}{2} + i \frac{2 ^{\frac{3}{8}} 2 + 2 + 2}{2}, z = - \frac{2 ^{\frac{3}{8}} 2 + 2 + 2}{2} - i \frac{2 ^{\frac{3}{8}} 2 - 2 + 2}{2}, z = \frac{2 ^{\frac{3}{8}} 2 - 2 + 2}{2} - i \frac{2 ^{\frac{3}{8}} 2 + 2 + 2}{2}