# How do you find all the real and complex roots of  x^4=1?

May 5, 2016

Real roots of the equation are $1$ and $- 1$

and complex roots are $i$ and $- i$.

#### Explanation:

As ${x}^{4} = 1$, $\left({x}^{4} - 1\right) = 0$

Using ${a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$ above is equal to

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

or $\left({x}^{2} - {i}^{2}\right) \left({x}^{2} - 1\right)$, (as ${i}^{2} = - 1$)

or $\left(x + i\right) \left(x - i\right) \left(x + 1\right) \left(x - 1\right) = 0$

Hence real roots of the equation are $1$ and $- 1$

and complex roots are $i$ and $- i$.