z^4=128+128sqrt(3)i

解

z^{4} = 128 + 1283 i

z = (6 + 2) + i (- 2 + 6), z = (2 - 6) + i (2 + 6), z = (- 6 - 2) + i (2 - 6), z = (6 - 2) + i (- 2 - 6)

解答ステップ

z^{4} = 128 + 1283 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 = 4, a = 128 + 1283 i

∣ a ∣ = 256

ar g (a) = \frac{π}{3}

z = 4256 (cos (\frac{\frac{π}{3} + 2 \cdot 0 π}{4}) + i sin (\frac{\frac{π}{3} + 2 \cdot 0 π}{4})), z = 4256 (cos (\frac{\frac{π}{3} + 2 \cdot 1 π}{4}) + i sin (\frac{\frac{π}{3} + 2 \cdot 1 π}{4})), z = 4256 (cos (\frac{\frac{π}{3} + 2 \cdot 2 π}{4}) + i sin (\frac{\frac{π}{3} + 2 \cdot 2 π}{4})), z = 4256 (cos (\frac{\frac{π}{3} + 2 \cdot 3 π}{4}) + i sin (\frac{\frac{π}{3} + 2 \cdot 3 π}{4}))

簡素化

4256 (cos (\frac{\frac{π}{3} + 2 \cdot 0 π}{4}) + i sin (\frac{\frac{π}{3} + 2 \cdot 0 π}{4})) : (6 + 2) + i (- 2 + 6)

簡素化

4256 (cos (\frac{\frac{π}{3} + 2 \cdot 1 π}{4}) + i sin (\frac{\frac{π}{3} + 2 \cdot 1 π}{4})) : (2 - 6) + i (2 + 6)

簡素化

4256 (cos (\frac{\frac{π}{3} + 2 \cdot 2 π}{4}) + i sin (\frac{\frac{π}{3} + 2 \cdot 2 π}{4})) : (- 6 - 2) + i (2 - 6)

簡素化

4256 (cos (\frac{\frac{π}{3} + 2 \cdot 3 π}{4}) + i sin (\frac{\frac{π}{3} + 2 \cdot 3 π}{4})) : (6 - 2) + i (- 2 - 6)

z = (6 + 2) + i (- 2 + 6), z = (2 - 6) + i (2 + 6), z = (- 6 - 2) + i (2 - 6), z = (6 - 2) + i (- 2 - 6)