x^3-27i=0

解

x^{3} - 27 i = 0

x = \frac{3 3}{2} + i \frac{3}{2}, x = - \frac{3 3}{2} + i \frac{3}{2}, x = - 3 i

解答ステップ

x^{3} - 27 i = 0

27 i

を右側に移動します

x^{3} = 27 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 = 27 i

∣ a ∣ = 27

ar g (a) = \frac{π}{2}

x = 327 (cos (\frac{\frac{π}{2} + 2 \cdot 0 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \cdot 0 π}{3})), x = 327 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{3})), x = 327 (cos (\frac{\frac{π}{2} + 2 \cdot 2 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \cdot 2 π}{3}))

簡素化

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

簡素化

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

簡素化

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

x = \frac{3 3}{2} + i \frac{3}{2}, x = - \frac{3 3}{2} + i \frac{3}{2}, x = - 3 i