vereinfachen 1/((iw+1)^4)

Lösung

vereinfachen

\frac{1}{( i w + 1 ) ^{4}}

\frac{w ^{4} - 6 w ^{2} + 1}{w ^{8} + 4 w ^{6} + 6 w ^{4} + 4 w ^{2} + 1} + \frac{4 w ^{3} - 4 w}{w ^{8} + 4 w ^{6} + 6 w ^{4} + 4 w ^{2} + 1} i

Schritte zur Lösung

\frac{1}{( i w + 1 ) ^{4}}

Schreibe

(i w + 1)^{4}

um:

w^{4} - 4 i w^{3} - 6 w^{2} + 4 i w + 1

= \frac{1}{w ^{4} - 4 i w ^{3} - 6 w ^{2} + 4 i w + 1}

Schreibe

w^{4} - 4 i w^{3} - 6 w^{2} + 4 i w + 1

in der Standard komplexen Form um:

(w^{4} - 6 w^{2} + 1) + (- 4 w^{3} + 4 w) i

= \frac{1}{( w ^{4} - 6 w ^{2} + 1 ) + ( - 4 w ^{3} + 4 w ) i}

Multipliziere mit dem Konjugat

\frac{( w ^{4} - 6 w ^{2} + 1 ) + ( 4 w ^{3} - 4 w ) i}{( w ^{4} - 6 w ^{2} + 1 ) + ( 4 w ^{3} - 4 w ) i}

= \frac{1 \cdot ( ( w ^{4} - 6 w ^{2} + 1 ) + ( 4 w ^{3} - 4 w ) i )}{( ( w ^{4} - 6 w ^{2} + 1 ) + ( - 4 w ^{3} + 4 w ) i ) ( ( w ^{4} - 6 w ^{2} + 1 ) + ( 4 w ^{3} - 4 w ) i )}

Vereinfache

\frac{1 \cdot ( ( w ^{4} - 6 w ^{2} + 1 ) + ( 4 w ^{3} - 4 w ) i )}{( ( w ^{4} - 6 w ^{2} + 1 ) + ( - 4 w ^{3} + 4 w ) i ) ( ( w ^{4} - 6 w ^{2} + 1 ) + ( 4 w ^{3} - 4 w ) i )} : \frac{w ^{4} - 6 w ^{2} + 1 + i ( 4 w ^{3} - 4 w )}{w ^{8} + 4 w ^{6} + 6 w ^{4} + 4 w ^{2} + 1}

= \frac{w ^{4} - 6 w ^{2} + 1 + i ( 4 w ^{3} - 4 w )}{w ^{8} + 4 w ^{6} + 6 w ^{4} + 4 w ^{2} + 1}

Rewrite in standard complex form:

\frac{w ^{4} - 6 w ^{2} + 1}{w ^{8} + 4 w ^{6} + 6 w ^{4} + 4 w ^{2} + 1} + \frac{4 w ^{3} - 4 w}{w ^{8} + 4 w ^{6} + 6 w ^{4} + 4 w ^{2} + 1} i

= \frac{w ^{4} - 6 w ^{2} + 1}{w ^{8} + 4 w ^{6} + 6 w ^{4} + 4 w ^{2} + 1} + \frac{4 w ^{3} - 4 w}{w ^{8} + 4 w ^{6} + 6 w ^{4} + 4 w ^{2} + 1} i

e x p an d (4 + 9) d y - 9 - 4 y^{2} = 0

s im pl i f y 2 (3 \cdot 3^{\frac{1}{3}} + 3 \cdot 6^{\frac{1}{3}})

e x p an d \frac{π e ^{- s} + π}{( s ^{2} + π ^{2} ) ( s ^{2} + 4 )}

s im pl i f y - 4 (5 - 3)

e x p an d \frac{1}{( x + 1 ) ln ( x + 1 )}