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受欢迎的代数 >

z^4=-4-4i

解答

z^{4} = - 4 - 4 i

解答

z = \frac{2 ^{\frac{5}{8}} 2 + 2 - 2}{2} - i \frac{2 ^{\frac{5}{8}} 2 - 2 - 2}{2}, z = \frac{2 ^{\frac{5}{8}} 2 - 2 - 2}{2} + i \frac{2 ^{\frac{5}{8}} 2 + 2 - 2}{2}, z = - \frac{2 ^{\frac{5}{8}} 2 + 2 - 2}{2} + i \frac{2 ^{\frac{5}{8}} 2 - 2 - 2}{2}, z = - \frac{2 ^{\frac{5}{8}} 2 - 2 - 2}{2} - i \frac{2 ^{\frac{5}{8}} 2 + 2 - 2}{2}

求解步骤

z^{4} = - 4 - 4 i

For

z^{n} = a

the solutions are

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

对于

n = 4, a = - 4 - 4 i

∣ a ∣ = 42

ar g (a) = \frac{π}{4} - π

z = 442 (cos (\frac{\frac{π}{4} - π + 2 \cdot 0 π}{4}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 0 π}{4})), z = 442 (cos (\frac{\frac{π}{4} - π + 2 \cdot 1 π}{4}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 1 π}{4})), z = 442 (cos (\frac{\frac{π}{4} - π + 2 \cdot 2 π}{4}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 2 π}{4})), z = 442 (cos (\frac{\frac{π}{4} - π + 2 \cdot 3 π}{4}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 3 π}{4}))

化简

442 (cos (\frac{\frac{π}{4} - π + 2 \cdot 0 π}{4}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 0 π}{4})) : \frac{2 ^{\frac{5}{8}} 2 + 2 - 2}{2} - i \frac{2 ^{\frac{5}{8}} 2 - 2 - 2}{2}

化简

442 (cos (\frac{\frac{π}{4} - π + 2 \cdot 1 π}{4}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 1 π}{4})) : \frac{2 ^{\frac{5}{8}} 2 - 2 - 2}{2} + i \frac{2 ^{\frac{5}{8}} 2 + 2 - 2}{2}

化简

442 (cos (\frac{\frac{π}{4} - π + 2 \cdot 2 π}{4}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 2 π}{4})) : - \frac{2 ^{\frac{5}{8}} 2 + 2 - 2}{2} + i \frac{2 ^{\frac{5}{8}} 2 - 2 - 2}{2}

化简

442 (cos (\frac{\frac{π}{4} - π + 2 \cdot 3 π}{4}) + i sin (\frac{\frac{π}{4} - π + 2 \cdot 3 π}{4})) : - \frac{2 ^{\frac{5}{8}} 2 - 2 - 2}{2} - i \frac{2 ^{\frac{5}{8}} 2 + 2 - 2}{2}

z = \frac{2 ^{\frac{5}{8}} 2 + 2 - 2}{2} - i \frac{2 ^{\frac{5}{8}} 2 - 2 - 2}{2}, z = \frac{2 ^{\frac{5}{8}} 2 - 2 - 2}{2} + i \frac{2 ^{\frac{5}{8}} 2 + 2 - 2}{2}, z = - \frac{2 ^{\frac{5}{8}} 2 + 2 - 2}{2} + i \frac{2 ^{\frac{5}{8}} 2 - 2 - 2}{2}, z = - \frac{2 ^{\frac{5}{8}} 2 - 2 - 2}{2} - i \frac{2 ^{\frac{5}{8}} 2 + 2 - 2}{2}

流行的例子

0 = x^{4} + 4 x^{2} - 5

2 x^{3} = 28

2 x^{3} = 10

11 = r^{11}

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