Answer
The possible solutions for $x$ are:
$cos~67.5^{\circ}+i~sin~67.5^{\circ}$
$cos~157.5^{\circ}+i~sin~157.5^{\circ}$
$cos~247.5^{\circ}+i~sin~247.5^{\circ}$
$cos~337.5^{\circ}+i~sin~337.5^{\circ}$
Work Step by Step
$x^4+i = 0$
$x^4 = -i$
$x = (-i)^{1/4}$
We need to find the fourth roots of $-i$.
Let $z = 0 - i$
$z = cos~270^{\circ}+i~sin~270^{\circ}$
$r = 1$ and $\theta = 270^{\circ}$
We can use this equation to find the fourth 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:
$z^{1/4} = 1^{1/4}~[cos(\frac{270^{\circ}}{4}+\frac{(360^{\circ})(0)}{4})+i~sin(\frac{270^{\circ}}{4}+\frac{(360^{\circ})(0)}{4})]$
$z^{1/4} = 1~[cos~67.5^{\circ}+i~sin~67.5^{\circ}]$
$z^{1/4} = cos~67.5^{\circ}+i~sin~67.5^{\circ}$
When k = 1:
$z^{1/4} = 1^{1/4}~[cos(\frac{270^{\circ}}{4}+\frac{(360^{\circ})(1)}{4})+i~sin(\frac{270^{\circ}}{4}+\frac{(360^{\circ})(1)}{4})]$
$z^{1/4} = 1~[cos~157.5^{\circ}+i~sin~157.5^{\circ}]$
$z^{1/4} = cos~157.5^{\circ}+i~sin~157.5^{\circ}$
When k = 2:
$z^{1/4} = 1^{1/4}~[cos(\frac{270^{\circ}}{4}+\frac{(360^{\circ})(2)}{4})+i~sin(\frac{270^{\circ}}{4}+\frac{(360^{\circ})(2)}{4})]$
$z^{1/4} = 1~[cos~247.5^{\circ}+i~sin~247.5^{\circ}]$
$z^{1/4} = cos~247.5^{\circ}+i~sin~247.5^{\circ}$
When k = 3:
$z^{1/4} = 1^{1/4}~[cos(\frac{270^{\circ}}{4}+\frac{(360^{\circ})(3)}{4})+i~sin(\frac{270^{\circ}}{4}+\frac{(360^{\circ})(3)}{4})]$
$z^{1/4} = 1~[cos~337.5^{\circ}+i~sin~337.5^{\circ}]$
$z^{1/4} = cos~337.5^{\circ}+i~sin~337.5^{\circ}$