2.2 Cube Roots of Unity and Their Properties

2.2 Cube Roots of Unity and Their Properties.

2.2.1 The cube roots of unity.

Let a number $x$ be the cube root of unity, i.e., $x = (1)^{1 / 3}$ or $x^{3} = 1$ $\Rightarrow x^3-1=0$$\Rightarrow x^{3} - 1 = 0$ $(x^{3}) - (1)^{3} = 0$ $(x - 1)(x^{2} + x + 1) = 0$ [using $a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2}]$

Either $x - 1 = 0$ or $x^{2} + x + 1 = 0$

From $x - 1 = 0$ , we get $x = 1$ .