x^4+i=0

解

x^{4} + i = 0

x = \frac{2 + 2}{2} - i \frac{2 - 2}{2}, x = \frac{2 - 2}{2} + i \frac{2 + 2}{2}, x = - \frac{2 + 2}{2} + i \frac{2 - 2}{2}, x = - \frac{2 - 2}{2} - i \frac{2 + 2}{2}

解答ステップ

x^{4} + i = 0

i

を右側に移動します

x^{4} = - 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 = - i

∣ a ∣ = 1

ar g (a) = - \frac{π}{2}

x = 41 (cos (\frac{- \frac{π}{2} + 2 \cdot 0 π}{4}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 0 π}{4})), x = 41 (cos (\frac{- \frac{π}{2} + 2 \cdot 1 π}{4}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 1 π}{4})), x = 41 (cos (\frac{- \frac{π}{2} + 2 \cdot 2 π}{4}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 2 π}{4})), x = 41 (cos (\frac{- \frac{π}{2} + 2 \cdot 3 π}{4}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 3 π}{4}))

簡素化

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

簡素化

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

簡素化

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

簡素化

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

x = \frac{2 + 2}{2} - i \frac{2 - 2}{2}, x = \frac{2 - 2}{2} + i \frac{2 + 2}{2}, x = - \frac{2 + 2}{2} + i \frac{2 - 2}{2}, x = - \frac{2 - 2}{2} - i \frac{2 + 2}{2}