x^6=36sqrt(3)+36i

解

x^{6} = 363 + 36 i

x = 672 cos (\frac{π}{36}) + 672 i sin (\frac{π}{36}), x = 672 cos (\frac{13 π}{36}) + 672 i sin (\frac{13 π}{36}), x = 672 cos (\frac{25 π}{36}) + 672 i sin (\frac{25 π}{36}), x = 672 cos (\frac{37 π}{36}) + 672 i sin (\frac{37 π}{36}), x = 672 cos (\frac{49 π}{36}) + 672 i sin (\frac{49 π}{36}), x = 672 cos (\frac{61 π}{36}) + 672 i sin (\frac{61 π}{36})

解答ステップ

x^{6} = 363 + 36 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 = 6, a = 363 + 36 i

∣ a ∣ = 72

ar g (a) = \frac{π}{6}

x = 672 (cos (\frac{\frac{π}{6} + 2 \cdot 0 π}{6}) + i sin (\frac{\frac{π}{6} + 2 \cdot 0 π}{6})), x = 672 (cos (\frac{\frac{π}{6} + 2 \cdot 1 π}{6}) + i sin (\frac{\frac{π}{6} + 2 \cdot 1 π}{6})), x = 672 (cos (\frac{\frac{π}{6} + 2 \cdot 2 π}{6}) + i sin (\frac{\frac{π}{6} + 2 \cdot 2 π}{6})), x = 672 (cos (\frac{\frac{π}{6} + 2 \cdot 3 π}{6}) + i sin (\frac{\frac{π}{6} + 2 \cdot 3 π}{6})), x = 672 (cos (\frac{\frac{π}{6} + 2 \cdot 4 π}{6}) + i sin (\frac{\frac{π}{6} + 2 \cdot 4 π}{6})), x = 672 (cos (\frac{\frac{π}{6} + 2 \cdot 5 π}{6}) + i sin (\frac{\frac{π}{6} + 2 \cdot 5 π}{6}))

簡素化

672 (cos (\frac{\frac{π}{6} + 2 \cdot 0 π}{6}) + i sin (\frac{\frac{π}{6} + 2 \cdot 0 π}{6})) : 672 cos (\frac{π}{36}) + 672 i sin (\frac{π}{36})

簡素化

672 (cos (\frac{\frac{π}{6} + 2 \cdot 1 π}{6}) + i sin (\frac{\frac{π}{6} + 2 \cdot 1 π}{6})) : 672 cos (\frac{13 π}{36}) + 672 i sin (\frac{13 π}{36})

簡素化

672 (cos (\frac{\frac{π}{6} + 2 \cdot 2 π}{6}) + i sin (\frac{\frac{π}{6} + 2 \cdot 2 π}{6})) : 672 cos (\frac{25 π}{36}) + 672 i sin (\frac{25 π}{36})

簡素化

672 (cos (\frac{\frac{π}{6} + 2 \cdot 3 π}{6}) + i sin (\frac{\frac{π}{6} + 2 \cdot 3 π}{6})) : 672 cos (\frac{37 π}{36}) + 672 i sin (\frac{37 π}{36})

簡素化

672 (cos (\frac{\frac{π}{6} + 2 \cdot 4 π}{6}) + i sin (\frac{\frac{π}{6} + 2 \cdot 4 π}{6})) : 672 cos (\frac{49 π}{36}) + 672 i sin (\frac{49 π}{36})

簡素化

672 (cos (\frac{\frac{π}{6} + 2 \cdot 5 π}{6}) + i sin (\frac{\frac{π}{6} + 2 \cdot 5 π}{6})) : 672 cos (\frac{61 π}{36}) + 672 i sin (\frac{61 π}{36})

x = 672 cos (\frac{π}{36}) + 672 i sin (\frac{π}{36}), x = 672 cos (\frac{13 π}{36}) + 672 i sin (\frac{13 π}{36}), x = 672 cos (\frac{25 π}{36}) + 672 i sin (\frac{25 π}{36}), x = 672 cos (\frac{37 π}{36}) + 672 i sin (\frac{37 π}{36}), x = 672 cos (\frac{49 π}{36}) + 672 i sin (\frac{49 π}{36}), x = 672 cos (\frac{61 π}{36}) + 672 i sin (\frac{61 π}{36})