z^4=8sqrt(2)+8sqrt(2)i

解

z^{4} = 82 + 82 i

z = 2 + 2 + 2 + i 2 - 2 + 2, z = - 2 - 2 + 2 + i 2 + 2 + 2, z = - 2 + 2 + 2 - i 2 - 2 + 2, z = 2 - 2 + 2 - i 2 + 2 + 2

解答ステップ

z^{4} = 82 + 82 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 = 82 + 82 i

∣ a ∣ = 16

ar g (a) = \frac{π}{4}

z = 416 (cos (\frac{\frac{π}{4} + 2 \cdot 0 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 0 π}{4})), z = 416 (cos (\frac{\frac{π}{4} + 2 \cdot 1 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 1 π}{4})), z = 416 (cos (\frac{\frac{π}{4} + 2 \cdot 2 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 2 π}{4})), z = 416 (cos (\frac{\frac{π}{4} + 2 \cdot 3 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 3 π}{4}))

簡素化

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

簡素化

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

簡素化

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

簡素化

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

z = 2 + 2 + 2 + i 2 - 2 + 2, z = - 2 - 2 + 2 + i 2 + 2 + 2, z = - 2 + 2 + 2 - i 2 - 2 + 2, z = 2 - 2 + 2 - i 2 + 2 + 2