Answer
The fifth roots of 1 are:
1
$0.309+0.951~i$
$-0.809+0.588~i$
$-0.809-0.588~i$
$0.309-0.951~i$
Work Step by Step
We can find the fifth roots of 1:
$\theta = 0^{\circ}$:
$1^{1/5} = cos~\theta^{\circ}+i~sin~\theta^{\circ}$
$1^{1/5} = cos~0^{\circ}+i~sin~0^{\circ}$
$1^{1/5} = 1+0i$
$\theta = 72^{\circ}$:
$1^{1/5} = cos~\theta^{\circ}+i~sin~\theta^{\circ}$
$1^{1/5} = cos~72^{\circ}+i~sin~72^{\circ}$
$1^{1/5} = 0.309+0.951~i$
$\theta = 144^{\circ}$:
$1^{1/5} = cos~\theta^{\circ}+i~sin~\theta^{\circ}$
$1^{1/5} = cos~144^{\circ}+i~sin~144^{\circ}$
$1^{1/5} = -0.809+0.588~i$
$\theta = 216^{\circ}$:
$1^{1/5} = cos~\theta^{\circ}+i~sin~\theta^{\circ}$
$1^{1/5} = cos~216^{\circ}+i~sin~216^{\circ}$
$1^{1/5} = -0.809-0.588~i$
$\theta = 288^{\circ}$:
$1^{1/5} = cos~\theta^{\circ}+i~sin~\theta^{\circ}$
$1^{1/5} = cos~288^{\circ}+i~sin~288^{\circ}$
$1^{1/5} = 0.309-0.951~i$
