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受欢迎的代数 >

solvefor z,z^5-32i=0

解答

求解

z, z^{5} - 32 i = 0

解答

z = \frac{5 + 5}{2} + i \frac{3 - 5}{2}, z = 2 i, z = - \frac{5 + 5}{2} + i \frac{3 - 5}{2}, z = - \frac{10 - 2 5}{2} + i \frac{- 1 - 5}{2}, z = \frac{10 - 2 5}{2} + i \frac{- 1 - 5}{2}

求解步骤

z^{5} - 32 i = 0

将

32 i

到右边

z^{5} = 32 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 = 5, a = 32 i

∣ a ∣ = 32

ar g (a) = \frac{π}{2}

z = 532 (cos (\frac{\frac{π}{2} + 2 \cdot 0 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 0 π}{5})), z = 532 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{5})), z = 532 (cos (\frac{\frac{π}{2} + 2 \cdot 2 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 2 π}{5})), z = 532 (cos (\frac{\frac{π}{2} + 2 \cdot 3 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 3 π}{5})), z = 532 (cos (\frac{\frac{π}{2} + 2 \cdot 4 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 4 π}{5}))

化简

532 (cos (\frac{\frac{π}{2} + 2 \cdot 0 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 0 π}{5})) : \frac{5 + 5}{2} + i \frac{3 - 5}{2}

化简

532 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{5})) : 2 i

化简

532 (cos (\frac{\frac{π}{2} + 2 \cdot 2 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 2 π}{5})) : - \frac{5 + 5}{2} + i \frac{3 - 5}{2}

化简

532 (cos (\frac{\frac{π}{2} + 2 \cdot 3 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 3 π}{5})) : - \frac{10 - 2 5}{2} + i \frac{- 1 - 5}{2}

化简

532 (cos (\frac{\frac{π}{2} + 2 \cdot 4 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 4 π}{5})) : \frac{10 - 2 5}{2} + i \frac{- 1 - 5}{2}

z = \frac{5 + 5}{2} + i \frac{3 - 5}{2}, z = 2 i, z = - \frac{5 + 5}{2} + i \frac{3 - 5}{2}, z = - \frac{10 - 2 5}{2} + i \frac{- 1 - 5}{2}, z = \frac{10 - 2 5}{2} + i \frac{- 1 - 5}{2}

流行的例子

2(2x-5)-x^2+17x=16x^4

2 (2 x - 5) - x^{2} + 17 x = 16 x^{4}

2 x^{3} - 9 x^{2} + 10 = 0

(x^{4} - 1) = 0

x^3-6x^2+6x-5=0

x^{3} - 6 x^{2} + 6 x - 5 = 0

x^3-x^2-75/8 =0

x^{3} - x^{2} - \frac{75}{8} = 0