Question

If w be cube roots of unity then w^7+w^8+w^9=

Answer 1

Answer:

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Step-by-step explanation:

$to \: find : \\ ω {}^{7} + ω {}^{8} + ω {}^{9} \\ \\ so \: we \: know \: that \\ whenever \: ω \: is \: cube \: root \: of \: unity \\ \\ ω {}^{3} = 1 \\ ω + ω {}^{2} = - 1$

$thus \: then \\ \\ ω {}^{7} + ω {}^{8} + ω {}^{9} \\ \\ = (ω {}^{6} \times ω) + (ω {}^{6} \times ω {}^{2} ) + (ω {}^{3} ) {}^{3} \\ \\ =( (ω {}^{3} ) {}^{2} \times ω) + ((ω {}^{3} ) {}^{2} \times ω {}^{2} ) + (1) {}^{3} \\ \\ =( (1) {}^{2} \times ω) + ((1) {}^{2} \times ω {}^{2} ) + 1 \\ \\ = ω + ω {}^{2} + 1 \\ \\ = - 1 + 1 \\ \\ = 0$