Answer
The five solutions for the value of $x$ are:
$cos~18^{\circ}+i~sin~18^{\circ}$
$cos~90^{\circ}+i~sin~90^{\circ}$
$cos~162^{\circ}+i~sin~162^{\circ}$
$cos~234^{\circ}+i~sin~234^{\circ}$
$cos~306^{\circ}+i~sin~306^{\circ}$
Work Step by Step
$x^5-i = 0$
$x^5 = i$
$x = (i)^{1/5}$
We need to find the fifth roots of $i$.
Let $z = 0 + i$
$z = cos~90^{\circ}+i~sin~90^{\circ}$
$r = 1$ and $\theta = 90^{\circ}$
We can use this equation to find the fifth roots:
$z^{1/n} = r^{1/n}~[cos(\frac{\theta}{n}+\frac{360^{\circ}~k}{n})+i~sin(\frac{\theta}{n}+\frac{360^{\circ}~k}{n})]$, where $k \in \{0, 1, 2,...,n-1\}$
When k = 0:
$z^{1/5} = 1^{1/5}~[cos(\frac{90^{\circ}}{5}+\frac{(360^{\circ})(0)}{5})+i~sin(\frac{90^{\circ}}{5}+\frac{(360^{\circ})(0)}{5})]$
$z^{1/5} = 1~[cos~18^{\circ}+i~sin~18^{\circ}]$
$z^{1/5} = cos~18^{\circ}+i~sin~18^{\circ}$
When k = 1:
$z^{1/5} = 1^{1/5}~[cos(\frac{90^{\circ}}{5}+\frac{(360^{\circ})(1)}{5})+i~sin(\frac{90^{\circ}}{5}+\frac{(360^{\circ})(1)}{5})]$
$z^{1/5} = 1~[cos~90^{\circ}+i~sin~90^{\circ}]$
$z^{1/5} = cos~90^{\circ}+i~sin~90^{\circ}$
When k = 2:
$z^{1/5} = 1^{1/5}~[cos(\frac{90^{\circ}}{5}+\frac{(360^{\circ})(2)}{5})+i~sin(\frac{90^{\circ}}{5}+\frac{(360^{\circ})(2)}{5})]$
$z^{1/5} = 1~[cos~162^{\circ}+i~sin~162^{\circ}]$
$z^{1/5} = cos~162^{\circ}+i~sin~162^{\circ}$
When k = 3:
$z^{1/5} = 1^{1/5}~[cos(\frac{90^{\circ}}{5}+\frac{(360^{\circ})(3)}{5})+i~sin(\frac{90^{\circ}}{5}+\frac{(360^{\circ})(3)}{5})]$
$z^{1/5} = 1~[cos~234^{\circ}+i~sin~234^{\circ}]$
$z^{1/5} = cos~234^{\circ}+i~sin~234^{\circ}$
When k = 4:
$z^{1/5} = 1^{1/5}~[cos(\frac{90^{\circ}}{5}+\frac{(360^{\circ})(4)}{5})+i~sin(\frac{90^{\circ}}{5}+\frac{(360^{\circ})(4)}{5})]$
$z^{1/5} = 1~[cos~306^{\circ}+i~sin~306^{\circ}]$
$z^{1/5} = cos~306^{\circ}+i~sin~306^{\circ}$