# How do you find three cube roots of 1?

Jun 26, 2017

Three cube roots of $1$ are

$1$, $\frac{- 1 - i \sqrt{3}}{2}$ and $\frac{- 1 + i \sqrt{3}}{2}$

#### Explanation:

Let $x = \sqrt[3]{1}$, then ${x}^{3} = 1$ or

${x}^{3} - 1 = 0$

or $\left(x - 1\right) \left({x}^{2} + x + 1\right) = 0$

Hence either $x = 1$

or ${x}^{2} + x + 1 = 0$ and using quadratic formula

$x = \frac{- 1 \pm \sqrt{1 - 4 \times 1 \times 1}}{2}$

= $\frac{- 1 \pm \sqrt{- 3}}{2}$

= $\frac{- 1 \pm i \sqrt{3}}{2}$

Hence three cube roots of $1$ are

$1$, $\frac{- 1 - i \sqrt{3}}{2}$ and $\frac{- 1 + i \sqrt{3}}{2}$