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Beliebt Algebra >

vereinfachen (1+it+t^2)^3

Lösung

vereinfachen

(1 + i t + t^{2})^{3}

Lösung

(t^{6} + 1) + (3 t^{5} + 5 t^{3} + 3 t) i

Schritte zur Lösung

(1 + i t + t^{2})^{3}

Wende Exponentenregel an:

a^{b + c} = a^{b} \cdot a^{c}

(1 + i t + t^{2})^{3} = (1 + i t + t^{2}) (1 + i t + t^{2})^{2}

= (1 + i t + t^{2}) (1 + i t + t^{2})^{2}

Wende Exponentenregel an:

a^{b + c} = a^{b} \cdot a^{c}

(1 + i t + t^{2})^{2} = (1 + i t + t^{2}) (1 + i t + t^{2})

= (1 + i t + t^{2}) (1 + i t + t^{2}) (1 + i t + t^{2})

Schreibe

(1 + i t + t^{2}) (1 + i t + t^{2})

um:

t^{4} + t^{2} + 2 t i + 2 t^{3} i + 1

= (t^{4} + t^{2} + 2 t i + 2 t^{3} i + 1) (1 + i t + t^{2})

Setze Klammern

= t^{4} \cdot 1 + t^{4} i t + t^{4} t^{2} + t^{2} \cdot 1 + t^{2} i t + t^{2} t^{2} + 2 t i 1 + 2 t ii t + 2 t i t^{2} + 2 t^{3} i 1 + 2 t^{3} ii t + 2 t^{3} i t^{2} + 1 \cdot 1 + 1 i t + 1 \cdot t^{2}

Vereinfache

t^{4} \cdot 1 + t^{4} i t + t^{4} t^{2} + t^{2} \cdot 1 + t^{2} i t + t^{2} t^{2} + 2 t i 1 + 2 t ii t + 2 t i t^{2} + 2 t^{3} i 1 + 2 t^{3} ii t + 2 t^{3} i t^{2} + 1 \cdot 1 + 1 i t + 1 \cdot t^{2} : t^{6} + 3 t^{5} i + 5 t^{3} i + 3 t i + 1

= t^{6} + 3 t^{5} i + 5 t^{3} i + 3 t i + 1

Rewrite in standard complex form:

(t^{6} + 1) + (3 t^{5} + 5 t^{3} + 3 t) i

= (t^{6} + 1) + (3 t^{5} + 5 t^{3} + 3 t) i

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