erweitern (10^6K)/((iw+10^2)^3)

Lösung

erweitern

\frac{1 0 ^{6} K}{( i w + 1 0 ^{2} ) ^{3}}

\frac{1000000000000 K - 300000000 K w ^{2}}{100000 0 ^{2} + w ^{6} + 300000000 w ^{2} + 30000 w ^{4}} + i \frac{1000000 K w ^{3} - 30000000000 K w}{100000 0 ^{2} + w ^{6} + 300000000 w ^{2} + 30000 w ^{4}}

Schritte zur Lösung

\frac{1 0 ^{6} K}{( i w + 1 0 ^{2} ) ^{3}}

1 0^{2} = 100

= \frac{1 0 ^{6} K}{( i w + 100 ) ^{3}}

(i w + 100)^{3} = - i w^{3} - 300 w^{2} + 30000 i w + 1000000

= \frac{1 0 ^{6} K}{( - i w ^{3} - 300 w ^{2} + 30000 i w + 1000000 )}

Fasse zusammen

= \frac{1000000 K}{- i w ^{3} - 300 w ^{2} + 30000 i w + 1000000}

Rationalisiere

\frac{1000000 K}{- i w ^{3} - 300 w ^{2} + 30000 i w + 1000000} : \frac{1000000 K ( - 300 w ^{2} + i ( w ^{3} - 30000 w ) + 1000000 )}{w ^{6} + 30000 w ^{4} + 300000000 w ^{2} + 100000 0 ^{2}}

= \frac{1000000 K ( - 300 w ^{2} + i ( w ^{3} - 30000 w ) + 1000000 )}{w ^{6} + 30000 w ^{4} + 300000000 w ^{2} + 100000 0 ^{2}}

Schreibe

\frac{1000000 K ( - 300 w ^{2} + 1000000 + i ( w ^{3} - 30000 w ) )}{w ^{6} + 30000 w ^{4} + 300000000 w ^{2} + 100000 0 ^{2}}

in der Standard komplexen Form um:

\frac{- 300000000 K w ^{2} + 1000000000000 K}{w ^{6} + 30000 w ^{4} + 300000000 w ^{2} + 100000 0 ^{2}} + \frac{1000000 K w ^{3} - 30000000000 K w}{w ^{6} + 30000 w ^{4} + 300000000 w ^{2} + 100000 0 ^{2}} i

= \frac{- 300000000 K w ^{2} + 1000000000000 K}{w ^{6} + 30000 w ^{4} + 300000000 w ^{2} + 100000 0 ^{2}} + \frac{1000000 K w ^{3} - 30000000000 K w}{w ^{6} + 30000 w ^{4} + 300000000 w ^{2} + 100000 0 ^{2}} i

e x p an d \frac{( p - 11 ) r ( p - 6 )}{2 ( p ^{2} - 12 p + 36 - 5 r )}

s im pl i f y (n - 6) (n - \frac{1}{2})

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

e x p an d \frac{( X + 25 ) ( X + 3 )}{X + 5}

e x p an d (1000 sin (100 t) + 1) \cdot (0.1 cos (100 t))