erweitern ((a+bi)^6)/((a-bi)^2)

Lösung

erweitern

\frac{( a + bi ) ^{6}}{( a - bi ) ^{2}}

\frac{a ^{8} + b ^{8} - 28 a ^{2} b ^{6} - 28 a ^{6} b ^{2} + 70 a ^{4} b ^{4}}{a ^{4} + b ^{4} + 2 a ^{2} b ^{2}} + i \frac{56 a ^{3} b ^{5} - 56 a ^{5} b ^{3} + 8 a ^{7} b - 8 a b ^{7}}{a ^{4} + b ^{4} + 2 a ^{2} b ^{2}}

Schritte zur Lösung

\frac{( a + bi ) ^{6}}{( a - bi ) ^{2}}

(a - bi)^{2} = a^{2} - 2 iab - b^{2}

= \frac{( a + ib ) ^{6}}{a ^{2} - 2 iab - b ^{2}}

(a + bi)^{6} = a^{6} + 6 i a^{5} b - 15 a^{4} b^{2} - 20 i a^{3} b^{3} + 15 a^{2} b^{4} + 6 ia b^{5} - b^{6}

= \frac{a ^{6} + 6 i a ^{5} b - 15 a ^{4} b ^{2} - 20 i a ^{3} b ^{3} + 15 a ^{2} b ^{4} + 6 ia b ^{5} - b ^{6}}{a ^{2} - 2 iab - b ^{2}}

Rationalisiere

\frac{a ^{6} + 6 i a ^{5} b - 15 a ^{4} b ^{2} - 20 i a ^{3} b ^{3} + 15 a ^{2} b ^{4} + 6 ia b ^{5} - b ^{6}}{a ^{2} - 2 iab - b ^{2}} : \frac{a ^{8} - 28 a ^{6} b ^{2} + 70 a ^{4} b ^{4} - 28 a ^{2} b ^{6} + i ( 8 a ^{7} b - 56 a ^{5} b ^{3} + 56 a ^{3} b ^{5} - 8 a b ^{7} ) + b ^{8}}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}}

= \frac{a ^{8} - 28 a ^{6} b ^{2} + 70 a ^{4} b ^{4} - 28 a ^{2} b ^{6} + b ^{8} + i ( 8 a ^{7} b - 56 a ^{5} b ^{3} + 56 a ^{3} b ^{5} - 8 a b ^{7} )}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}}

Schreibe

\frac{a ^{8} - 28 a ^{6} b ^{2} + 70 a ^{4} b ^{4} - 28 a ^{2} b ^{6} + b ^{8} + i ( 8 a ^{7} b - 56 a ^{5} b ^{3} + 56 a ^{3} b ^{5} - 8 a b ^{7} )}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}}

in der Standard komplexen Form um:

\frac{a ^{8} - 28 a ^{6} b ^{2} + 70 a ^{4} b ^{4} - 28 a ^{2} b ^{6} + b ^{8}}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}} + \frac{8 a ^{7} b - 56 a ^{5} b ^{3} + 56 a ^{3} b ^{5} - 8 a b ^{7}}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}} i

= \frac{a ^{8} - 28 a ^{6} b ^{2} + 70 a ^{4} b ^{4} - 28 a ^{2} b ^{6} + b ^{8}}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}} + \frac{8 a ^{7} b - 56 a ^{5} b ^{3} + 56 a ^{3} b ^{5} - 8 a b ^{7}}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}} i

e x p an d (cos (ln (x)) + sin (ln (x)))^{2}

e x p an d (1 - \frac{1 - cos ( x π )}{2}) 1 0^{\frac{x}{2}}

e x p an d \frac{1}{8} e^{4 x} (- 4 e^{- 4 t} t - e^{- 4 t})

e x p an d 2 cos (5) (2 tan (5) x + 25)

e x p an d y^{2} d x + (x^{2} + x y) d y