展开 ((3+4i)^4)/((2-i)^{10)}

解答

展开

\frac{( 3 + 4 i ) ^{4}}{( 2 - i ) ^{10}}

- \frac{922077}{9765625} + \frac{1721764 i}{9765625}

求解步骤

\frac{( 3 + 4 i ) ^{4}}{( 2 - i ) ^{10}}

(2 - i)^{10} = - 237 + 3116 i

= \frac{( 3 + 4 i ) ^{4}}{( - 237 + 3116 i )}

(3 + 4 i)^{4} = - 527 - 336 i

= \frac{( - 527 - 336 i )}{( - 237 + 3116 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 = - 527, b = - 336, c = - 237, d = 3116

= \frac{( - 527 ( - 237 ) + ( - 336 ) \cdot 3116 ) + ( - 336 ( - 237 ) - ( - 527 ) \cdot 3116 ) i}{( - 237 ) ^{2} + 311 6 ^{2}}

整理后得

= \frac{- 922077 + 1721764 i}{9765625}

使用分式法则:

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

\frac{- 922077 + 1721764 i}{9765625} = - \frac{922077}{9765625} + \frac{1721764 i}{9765625}

= - \frac{922077}{9765625} + \frac{1721764 i}{9765625}

e x p an d \frac{( x + 1 ) ^{2} ( x - 3 )}{( x - 4 ) ^{3} ( x + 5 )}

s im pl i f y (x + 2 - 2) \cdot (x + 2 + 2)

e x p an d \frac{- 2 x ^{2} + 10 x + 4}{( x - 1 ) ( x ^{2} + 2 x + 9 )}

s im pl i f y (x - 2)^{2} (x + 2)

e x p an d - e^{2 x} (2 e^{- 2 x} - e^{- x})