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受欢迎的代数 >

z^8-16i=0

解答

z^{8} - 16 i = 0

解答

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

求解步骤

z^{8} - 16 i = 0

将

16 i

到右边

z^{8} = 16 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 = 8, a = 16 i

∣ a ∣ = 16

ar g (a) = \frac{π}{2}

z = 816 (cos (\frac{\frac{π}{2} + 2 \cdot 0 π}{8}) + i sin (\frac{\frac{π}{2} + 2 \cdot 0 π}{8})), z = 816 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{8}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{8})), z = 816 (cos (\frac{\frac{π}{2} + 2 \cdot 2 π}{8}) + i sin (\frac{\frac{π}{2} + 2 \cdot 2 π}{8})), z = 816 (cos (\frac{\frac{π}{2} + 2 \cdot 3 π}{8}) + i sin (\frac{\frac{π}{2} + 2 \cdot 3 π}{8})), z = 816 (cos (\frac{\frac{π}{2} + 2 \cdot 4 π}{8}) + i sin (\frac{\frac{π}{2} + 2 \cdot 4 π}{8})), z = 816 (cos (\frac{\frac{π}{2} + 2 \cdot 5 π}{8}) + i sin (\frac{\frac{π}{2} + 2 \cdot 5 π}{8})), z = 816 (cos (\frac{\frac{π}{2} + 2 \cdot 6 π}{8}) + i sin (\frac{\frac{π}{2} + 2 \cdot 6 π}{8})), z = 816 (cos (\frac{\frac{π}{2} + 2 \cdot 7 π}{8}) + i sin (\frac{\frac{π}{2} + 2 \cdot 7 π}{8}))

化简

816 (cos (\frac{\frac{π}{2} + 2 \cdot 0 π}{8}) + i sin (\frac{\frac{π}{2} + 2 \cdot 0 π}{8})) : \frac{2 + 2 + 2}{2} + i \frac{2 - 2 + 2}{2}

化简

816 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{8}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{8})) : \frac{2 - 2 - 2}{2} + i \frac{2 + 2 - 2}{2}

化简

816 (cos (\frac{\frac{π}{2} + 2 \cdot 2 π}{8}) + i sin (\frac{\frac{π}{2} + 2 \cdot 2 π}{8})) : - \frac{2 - 2 + 2}{2} + i \frac{2 + 2 + 2}{2}

化简

816 (cos (\frac{\frac{π}{2} + 2 \cdot 3 π}{8}) + i sin (\frac{\frac{π}{2} + 2 \cdot 3 π}{8})) : - \frac{2 + 2 - 2}{2} + i \frac{2 - 2 - 2}{2}

化简

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

化简

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

化简

816 (cos (\frac{\frac{π}{2} + 2 \cdot 6 π}{8}) + i sin (\frac{\frac{π}{2} + 2 \cdot 6 π}{8})) : \frac{2 - 2 + 2}{2} - i \frac{2 + 2 + 2}{2}

化简

816 (cos (\frac{\frac{π}{2} + 2 \cdot 7 π}{8}) + i sin (\frac{\frac{π}{2} + 2 \cdot 7 π}{8})) : \frac{2 + 2 - 2}{2} - i \frac{2 - 2 - 2}{2}

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

流行的例子

x^4-2937x^2+100=0

x^{4} - 2937 x^{2} + 100 = 0

x^{2} (x - 3) = 0

4x^3-12x^2=9x-27

4 x^{3} - 12 x^{2} = 9 x - 27

x^{3} - 4 x + 3 = 0

2x^3-8x^2-24x=ax(x+b)(x+c)

2 x^{3} - 8 x^{2} - 24 x = a x (x + b) (x + c)