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人気のある代数 >

z^6+2i=0

解

z^{6} + 2 i = 0

解

z = \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( 1 - 3 )}{4}, z = \frac{2 ^{\frac{2}{3}}}{2} + i \frac{2 ^{\frac{2}{3}}}{2}, z = - \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4}, z = \frac{2 ^{\frac{2}{3}} ( - 1 - 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4}, z = - \frac{2 ^{\frac{2}{3}}}{2} - i \frac{2 ^{\frac{2}{3}}}{2}, z = \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4} - i \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4}

解答ステップ

z^{6} + 2 i = 0

2 i

を右側に移動します

z^{6} = - 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 = 6, a = - 2 i

∣ a ∣ = 2

ar g (a) = - \frac{π}{2}

z = 62 (cos (\frac{- \frac{π}{2} + 2 \cdot 0 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 0 π}{6})), z = 62 (cos (\frac{- \frac{π}{2} + 2 \cdot 1 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 1 π}{6})), z = 62 (cos (\frac{- \frac{π}{2} + 2 \cdot 2 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 2 π}{6})), z = 62 (cos (\frac{- \frac{π}{2} + 2 \cdot 3 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 3 π}{6})), z = 62 (cos (\frac{- \frac{π}{2} + 2 \cdot 4 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 4 π}{6})), z = 62 (cos (\frac{- \frac{π}{2} + 2 \cdot 5 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 5 π}{6}))

簡素化

62 (cos (\frac{- \frac{π}{2} + 2 \cdot 0 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 0 π}{6})) : \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( 1 - 3 )}{4}

簡素化

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

簡素化

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

簡素化

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

簡素化

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

簡素化

62 (cos (\frac{- \frac{π}{2} + 2 \cdot 5 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 5 π}{6})) : \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4} - i \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4}

z = \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( 1 - 3 )}{4}, z = \frac{2 ^{\frac{2}{3}}}{2} + i \frac{2 ^{\frac{2}{3}}}{2}, z = - \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4}, z = \frac{2 ^{\frac{2}{3}} ( - 1 - 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4}, z = - \frac{2 ^{\frac{2}{3}}}{2} - i \frac{2 ^{\frac{2}{3}}}{2}, z = \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4} - i \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4}

人気の例

(x-1)(x-2)(x^2-3x)=24

(x - 1) (x - 2) (x^{2} - 3 x) = 24

x^{2} + 11 x^{24} = 0

- a^{3} - a^{2} - a = 3

(2 x + 1)^{4} = 81

x^{3} - 5 x^{2} = 36 x