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Beliebt Algebra >

vereinfachen (a-bi)/(1+(a+bi)^2)

Lösung

vereinfachen

\frac{a - bi}{1 + ( a + bi ) ^{2}}

Lösung

\frac{a + a ^{3} - 3 a b ^{2}}{a ^{4} + b ^{4} + 2 a ^{2} - 2 b ^{2} + 3 a ^{2} b ^{2} - b ^{2} a ^{2} + 1} + \frac{b ^{3} - 3 a ^{2} b - b}{a ^{4} + b ^{4} + 2 a ^{2} - 2 b ^{2} + 3 a ^{2} b ^{2} - b ^{2} a ^{2} + 1} i

Schritte zur Lösung

\frac{a - bi}{1 + ( a + bi ) ^{2}}

Schreibe

1 + (a + bi)^{2}

um:

1 + a^{2} + 2 abi - b^{2}

= \frac{a - bi}{1 + a ^{2} + 2 abi - b ^{2}}

Schreibe

1 + a^{2} + 2 abi - b^{2}

in der Standard komplexen Form um:

(1 + a^{2} - b^{2}) + 2 abi

= \frac{a - bi}{( 1 + a ^{2} - b ^{2} ) + 2 abi}

Multipliziere mit dem Konjugat

\frac{( 1 + a ^{2} - b ^{2} ) - 2 abi}{( 1 + a ^{2} - b ^{2} ) - 2 abi}

= \frac{( a - bi ) ( ( 1 + a ^{2} - b ^{2} ) - 2 abi )}{( ( 1 + a ^{2} - b ^{2} ) + 2 abi ) ( ( 1 + a ^{2} - b ^{2} ) - 2 abi )}

Vereinfache

\frac{( a - bi ) ( ( 1 + a ^{2} - b ^{2} ) - 2 abi )}{( ( 1 + a ^{2} - b ^{2} ) + 2 abi ) ( ( 1 + a ^{2} - b ^{2} ) - 2 abi )} : \frac{a + a ^{3} - 3 a b ^{2} + ( b ^{3} - 3 a ^{2} b - b ) i}{a ^{4} + b ^{4} + 2 a ^{2} - 2 b ^{2} + 3 a ^{2} b ^{2} - b ^{2} a ^{2} + 1}

= \frac{a + a ^{3} - 3 a b ^{2} + ( b ^{3} - 3 a ^{2} b - b ) i}{a ^{4} + b ^{4} + 2 a ^{2} - 2 b ^{2} + 3 a ^{2} b ^{2} - b ^{2} a ^{2} + 1}

Rewrite in standard complex form:

\frac{a + a ^{3} - 3 a b ^{2}}{a ^{4} + b ^{4} + 2 a ^{2} - 2 b ^{2} + 3 a ^{2} b ^{2} - b ^{2} a ^{2} + 1} + \frac{b ^{3} - 3 a ^{2} b - b}{a ^{4} + b ^{4} + 2 a ^{2} - 2 b ^{2} + 3 a ^{2} b ^{2} - b ^{2} a ^{2} + 1} i

= \frac{a + a ^{3} - 3 a b ^{2}}{a ^{4} + b ^{4} + 2 a ^{2} - 2 b ^{2} + 3 a ^{2} b ^{2} - b ^{2} a ^{2} + 1} + \frac{b ^{3} - 3 a ^{2} b - b}{a ^{4} + b ^{4} + 2 a ^{2} - 2 b ^{2} + 3 a ^{2} b ^{2} - b ^{2} a ^{2} + 1} i

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