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人気のある代数 >

-i=x^6

解

- i = x^{6}

解

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

解答ステップ

- i = x^{6}

辺を交換する

x^{6} = - 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 = - i

∣ a ∣ = 1

ar g (a) = - \frac{π}{2}

x = 61 (cos (\frac{- \frac{π}{2} + 2 \cdot 0 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 0 π}{6})), x = 61 (cos (\frac{- \frac{π}{2} + 2 \cdot 1 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 1 π}{6})), x = 61 (cos (\frac{- \frac{π}{2} + 2 \cdot 2 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 2 π}{6})), x = 61 (cos (\frac{- \frac{π}{2} + 2 \cdot 3 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 3 π}{6})), x = 61 (cos (\frac{- \frac{π}{2} + 2 \cdot 4 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 4 π}{6})), x = 61 (cos (\frac{- \frac{π}{2} + 2 \cdot 5 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 5 π}{6}))

簡素化

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

簡素化

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

簡素化

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

簡素化

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

簡素化

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

簡素化

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

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

人気の例

x^{3} - 2 x^{2} - x + 2 = 0

x^{3} + x^{2} = 0

z^{3} + i = 0

x^{4} - 2 x^{2} - 63 = 0

3 x - 2 = x^{8}