展開する ((8-2i)^{10})/((4+6i)^5)

解

展開する

\frac{( 8 - 2 i ) ^{10}}{( 4 + 6 i ) ^{5}}

\frac{13035060672}{371293} - i \frac{24424909472}{371293}

解答ステップ

\frac{( 8 - 2 i ) ^{10}}{( 4 + 6 i ) ^{5}}

(4 + 6 i)^{5} = 3904 - 19104 i

= \frac{( 8 - 2 i ) ^{10}}{3904 - 19104 i}

(8 - 2 i)^{10} = - 927506432 i + 8^{10} - 2193409024

= \frac{( - 927506432 i + 8 ^{10} - 2193409024 )}{3904 - 19104 i}

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 = 8^{10} - 2193409024, b = - 927506432, c = 3904, d = - 19104

= \frac{( ( 8 ^{10} - 2193409024 ) \cdot 3904 + ( - 927506432 ) ( - 19104 ) ) + ( - 927506432 \cdot 3904 - ( 8 ^{10} - 2193409024 ) ( - 19104 ) ) i}{390 4 ^{2} + ( - 19104 ) ^{2}}

改良

= \frac{( 8 ^{10} \cdot 3904 + 9156014047232 ) + ( 8 ^{10} \cdot 19104 - 4.55239 E 13 ) i}{380204032}

簡素化

\frac{( 8 ^{10} \cdot 3904 + 9156014047232 ) + ( 8 ^{10} \cdot 19104 - 4.55239 E 13 ) i}{380204032} : \frac{8 ^{10} \cdot 3904 + 9156014047232 + i ( 8 ^{10} \cdot 19104 - 4.55239 E 13 )}{380204032}

= \frac{8 ^{10} \cdot 3904 + 9156014047232 + i ( 8 ^{10} \cdot 19104 - 4.55239 E 13 )}{380204032}

拡張

8^{10} \cdot 3904 + 9156014047232 + i (8^{10} \cdot 19104 - 4.55239 E 13) : 8^{10} \cdot 3904 + 9156014047232 + 8^{10} \cdot 19104 i - 4.55239 E 13 i

= \frac{8 ^{10} \cdot 3904 + 9156014047232 + 8 ^{10} \cdot 19104 i - 4.55239 E 13 i}{380204032}

分数の規則を適用する:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{8 ^{10} \cdot 3904 + 9156014047232 + 8 ^{10} \cdot 19104 i - 4.55239 E 13 i}{380204032} = \frac{8 ^{10} \cdot 3904}{380204032} + \frac{9156014047232}{380204032} + \frac{8 ^{10} \cdot 19104 i}{380204032} - \frac{4.55239 E 13 i}{380204032}

= \frac{8 ^{10} \cdot 3904}{380204032} + \frac{9156014047232}{380204032} + \frac{8 ^{10} \cdot 19104 i}{380204032} - \frac{4.55239 E 13 i}{380204032}

\frac{8 ^{10} \cdot 3904}{380204032} = \frac{4093640704}{371293}

\frac{9156014047232}{380204032} = \frac{687801536}{28561}

\frac{8 ^{10} \cdot 19104 i}{380204032} = \frac{20031995904 i}{371293}

\frac{4.55239 E 13 i}{380204032} = \frac{3419761952 i}{28561}

= \frac{4093640704}{371293} + \frac{687801536}{28561} + \frac{20031995904 i}{371293} - \frac{3419761952 i}{28561}

標準的な複素数形式で

\frac{4093640704}{371293} + \frac{687801536}{28561} + \frac{20031995904 i}{371293} - \frac{3419761952 i}{28561}

を書き換える:

\frac{13035060672}{371293} - \frac{24424909472}{371293} i

= \frac{13035060672}{371293} - \frac{24424909472}{371293} i