(A) We know that $1 + \omega + \omega^2 = 0$,which implies $1 + \omega^2 = -\omega$. Substituting this into the expression: $(1 - 2\omega + \omega^2)^

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(A) We know that $1 + \omega + \omega^2 = 0$,which implies $1 + \omega^2 = -\omega$. Substituting this into the expression: $(1 - 2\omega + \omega^2)^6 = (1 + \omega^2 - 2\omega)^6$ $= (-\omega - 2\omega)^6$ $= (-3\omega)^6$ $= (-3)^6 \times \omega^6$ Since $\omega^3 = 1$,we have $\omega^6 = (\omega^3)^2 = 1^2 = 1$. Therefore,$(-3)^6 \times 1 = 729$.

Class 11 Mathematics 1 Complex Numbers and Quadratic Equations: Complex Numbers De Moivre's theorem and Roots of unity

Medium

If $1, \omega, \omega^2$ are the cube roots of unity,then $(1 - 2\omega + \omega^2)^6$ is equal to

A
$729$
B
$246$
C
$243$
D
$81$

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1 Complex Numbers and Quadratic Equations: Complex Numbers

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De Moivre's theorem and Roots of unity

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If $1, \omega, \omega^2$ are the cube roots of unity,then $(1 - 2\omega + \omega^2)^6$ is equal to

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If $1, \omega, \omega^2$ are the cube roots of unity,then $\frac{1}{1+2 \omega}+\frac{1}{2+\omega}-\frac{1}{1+\omega}=$

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