展开 ((2+4i)^4)/((6+5i)^4)

解答

展开

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

- \frac{117232}{13845841} + \frac{1483776 i}{13845841}

求解步骤

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

(6 + 5 i)^{4} = - 3479 + 1320 i

= \frac{( 2 + 4 i ) ^{4}}{( - 3479 + 1320 i )}

(2 + 4 i)^{4} = - 112 - 384 i

= \frac{( - 112 - 384 i )}{( - 3479 + 1320 i )}

使用复数算数法则:

\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 = - 112, b = - 384, c = - 3479, d = 1320

= \frac{( - 112 ( - 3479 ) + ( - 384 ) \cdot 1320 ) + ( - 384 ( - 3479 ) - ( - 112 ) \cdot 1320 ) i}{( - 3479 ) ^{2} + 132 0 ^{2}}

整理后得

= \frac{- 117232 + 1483776 i}{13845841}

使用分式法则:

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

\frac{- 117232 + 1483776 i}{13845841} = - \frac{117232}{13845841} + \frac{1483776 i}{13845841}

= - \frac{117232}{13845841} + \frac{1483776 i}{13845841}

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