z^3=(-5sqrt(3)+5i)^3

解

z^{3} = (- 53 + 5 i)^{3}

z = 53 + 5 i, z = - 53 + 5 i, z = - 10 i

解答ステップ

z^{3} = (- 53 + 5 i)^{3}

簡素化

(- 53 + 5 i)^{3} : 1000 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 = 3, a = 1000 i

∣ a ∣ = 1000

ar g (a) = \frac{π}{2}

z = 31000 (cos (\frac{\frac{π}{2} + 2 \cdot 0 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \cdot 0 π}{3})), z = 31000 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{3})), z = 31000 (cos (\frac{\frac{π}{2} + 2 \cdot 2 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \cdot 2 π}{3}))

簡素化

31000 (cos (\frac{\frac{π}{2} + 2 \cdot 0 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \cdot 0 π}{3})) : 53 + 5 i

簡素化

31000 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{3})) : - 53 + 5 i

簡素化

31000 (cos (\frac{\frac{π}{2} + 2 \cdot 2 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \cdot 2 π}{3})) : - 10 i

z = 53 + 5 i, z = - 53 + 5 i, z = - 10 i