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人気のある代数 >

z^6=i

解

z^{6} = i

解

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

解答ステップ

z^{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}

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

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}{2} + i \frac{2}{2}

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

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}{2} - i \frac{2}{2}

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

人気の例

solvefor t,Fd^2+3=t^3

so l v e f or t, F d^{2} + 3 = t^{3}

21x^2-18x-3x^3=0

21 x^{2} - 18 x - 3 x^{3} = 0

150x^2+0.38x^3-80=0

150 x^{2} + 0.38 x^{3} - 80 = 0

x^3-12x^2+32x=0

x^{3} - 12 x^{2} + 32 x = 0

8 x^{6} - 7 x^{2} = 0