x^4-64i=0

解

x^{4} - 64 i = 0

x = 4 + 22 + i 4 - 22, x = - 4 - 22 + i 4 + 22, x = - 4 + 22 - i 4 - 22, x = 4 - 22 - i 4 + 22

解答ステップ

x^{4} - 64 i = 0

64 i

を右側に移動します

x^{4} = 64 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 = 64 i

∣ a ∣ = 64

ar g (a) = \frac{π}{2}

x = 464 (cos (\frac{\frac{π}{2} + 2 \cdot 0 π}{4}) + i sin (\frac{\frac{π}{2} + 2 \cdot 0 π}{4})), x = 464 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{4}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{4})), x = 464 (cos (\frac{\frac{π}{2} + 2 \cdot 2 π}{4}) + i sin (\frac{\frac{π}{2} + 2 \cdot 2 π}{4})), x = 464 (cos (\frac{\frac{π}{2} + 2 \cdot 3 π}{4}) + i sin (\frac{\frac{π}{2} + 2 \cdot 3 π}{4}))

簡素化

464 (cos (\frac{\frac{π}{2} + 2 \cdot 0 π}{4}) + i sin (\frac{\frac{π}{2} + 2 \cdot 0 π}{4})) : 4 + 22 + i 4 - 22

簡素化

464 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{4}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{4})) : - 4 - 22 + i 4 + 22

簡素化

464 (cos (\frac{\frac{π}{2} + 2 \cdot 2 π}{4}) + i sin (\frac{\frac{π}{2} + 2 \cdot 2 π}{4})) : - 4 + 22 - i 4 - 22

簡素化

464 (cos (\frac{\frac{π}{2} + 2 \cdot 3 π}{4}) + i sin (\frac{\frac{π}{2} + 2 \cdot 3 π}{4})) : 4 - 22 - i 4 + 22

x = 4 + 22 + i 4 - 22, x = - 4 - 22 + i 4 + 22, x = - 4 + 22 - i 4 - 22, x = 4 - 22 - i 4 + 22