1 4 Roots Of Unity – Class 11TH Math

1.4 Three Cube Roots of Unity

Let $x$ be a cube root of unity. Then

x = 1^{1/3} \implies x^3 = 1 \implies x^3 - 1 = 0 \implies (x - 1)(x^2 + x + 1) = 0

Either $x - 1 = 0 \Rightarrow x = 1$ , or $x^2 + x + 1 = 0$ :

x = \dfrac{-1 \pm \sqrt{1 - 4}}{2} = \dfrac{-1 \pm \sqrt{3}\,i}{2} \quad (\because\, \sqrt{-1} = i)

Thus the three cube roots of unity are

1,\quad \dfrac{-1 + \sqrt{3}\,i}{2},\quad \dfrac{-1 - \sqrt{3}\,i}{2}

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Note. $\dfrac{-1 + \sqrt{3}\,i}{2}$ and $\dfrac{-1 - \sqrt{3}\,i}{2}$ are called the imaginary cube roots of unity.