展開する ((-sqrt(3)+2i)^5)/((-5sqrt(3)+15i)^7)

解

展開する

\frac{( - 3 + 2 i ) ^{5}}{( - 5 3 + 15 i ) ^{7}}

- \frac{60345000000}{40500000 0 ^{2} + 13500000 0 ^{2} \cdot 3} + i \frac{3375000000 3}{40500000 0 ^{2} + 13500000 0 ^{2} \cdot 3}

解答ステップ

\frac{( - 3 + 2 i ) ^{5}}{( - 5 3 + 15 i ) ^{7}}

(- 53 + 15 i)^{7} = 405000000 i - 1350000003

= \frac{( - 3 + 2 i ) ^{5}}{405000000 i - 135000000 3}

(- 3 + 2 i)^{5} = - 118 i + 313

= \frac{( - 118 i + 31 3 )}{405000000 i - 135000000 3}

複素数の算術規則を適用する:

\frac{a + bi}{c + d i} = \frac{( c - d i ) ( a + bi )}{( c - d i ) ( c + d i )} = \frac{( a c + b d ) + ( b c - a d ) i}{c ^{2} + d ^{2}}

a = 313, b = - 118, c = - 1350000003, d = 405000000

= \frac{( 31 3 ( - 135000000 3 ) + ( - 118 ) \cdot 405000000 ) + ( - 118 ( - 135000000 3 ) - 31 3 \cdot 405000000 ) i}{( - 135000000 3 ) ^{2} + 40500000 0 ^{2}}

改良

= \frac{- 60345000000 + 3375000000 3 i}{13500000 0 ^{2} \cdot 3 + 40500000 0 ^{2}}

標準的な複素数形式で

\frac{- 60345000000 + 3375000000 3 i}{13500000 0 ^{2} \cdot 3 + 40500000 0 ^{2}}

を書き換える:

- \frac{60345000000}{13500000 0 ^{2} \cdot 3 + 40500000 0 ^{2}} + \frac{3375000000 3}{13500000 0 ^{2} \cdot 3 + 40500000 0 ^{2}} i

= - \frac{60345000000}{13500000 0 ^{2} \cdot 3 + 40500000 0 ^{2}} + \frac{3375000000 3}{13500000 0 ^{2} \cdot 3 + 40500000 0 ^{2}} i