(B) The cube roots of unity are $1, \omega, \omega^2$,where $\omega = e^{i\frac{2\pi}{3}}$ and $\omega^2 = e^{i\frac{4\pi}{3}}$.Since $\omega^2$ is th

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The other imaginary cube root of unity

(B) The cube roots of unity are $1, \omega, \omega^2$,where $\omega = e^{i\frac{2\pi}{3}}$ and $\omega^2 = e^{i\frac{4\pi}{3}}$.Since $\omega^2$ is the square of $\omega$,and $(\omega^2)^2 = \omega^4 = \omega^3 \cdot \omega = 1 \cdot \omega = \omega$,it follows that the square of either imaginary cube root is the other imaginary cube root.

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

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The square of either of the two imaginary cube roots of unity is:

A
Real root of unity
B
The other imaginary cube root of unity
C
The sum of the two imaginary roots of unity
D
None of these

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Chapter

1 Complex Numbers and Quadratic Equations: Complex Numbers

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De Moivre's theorem and Roots of unity

If ${\left( {\frac{{1 + \cos \theta + i\sin \theta }}{{i + \sin \theta + i\cos \theta }}} \right)^4} = \cos n\theta + i\sin n\theta $,then $n$ is equal to

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If $\omega$ is a non-real cube root of unity and $x = \omega^2 - \omega - 3$,then the value of $x^4 + 6x^3 + 10x^2 - 12x - 19$ is

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If $\omega$ is a complex cube root of unity,then $(3 + 5\omega + 3\omega^2)^2 + (3 + 3\omega + 5\omega^2)^2 = $

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${\left( \frac{-1 + i\sqrt{3}}{2} \right)^{20}} + {\left( \frac{-1 - i\sqrt{3}}{2} \right)^{20}} = $

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If $z+\frac{1}{z}=1$,then $\frac{\left(z^{20}+1\right)\left(z^{40}+1\right)\left(z^{60}+1\right)}{z^{60}}=$

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The square of either of the two imaginary cube roots of unity is:

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