Answer
The four possible value for $x$ are:
$2~(cos~45^{\circ}+i~sin~45^{\circ})$
$2~(cos~135^{\circ}+i~sin~135^{\circ})$
$2~(cos~225^{\circ}+i~sin~225^{\circ})$
$2~(cos~315^{\circ}+i~sin~315^{\circ})$
Work Step by Step
$x^4+16 = 0$
$x^4 = -16$
$x = (-16)^{1/4}$
$x = [16(-1+0~i)]^{1/4}$
Let $z = 16(-1 + 0~i)$
$z = 16(cos~180^{\circ}+i~sin~180^{\circ})$
$r = 16$ and $\theta = 180^{\circ}$
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:
$z^{1/4} = 16^{1/4}~[cos(\frac{180^{\circ}}{4}+\frac{(360^{\circ})(0)}{4})+i~sin(\frac{180^{\circ}}{4}+\frac{(360^{\circ})(0)}{4})]$
$z^{1/4} = 2~(cos~45^{\circ}+i~sin~45^{\circ})$
When k = 1:
$z^{1/4} = 16^{1/4}~[cos(\frac{180^{\circ}}{4}+\frac{(360^{\circ})(1)}{4})+i~sin(\frac{180^{\circ}}{4}+\frac{(360^{\circ})(1)}{4})]$
$z^{1/4} = 2~(cos~135^{\circ}+i~sin~135^{\circ})$
When k = 2:
$z^{1/4} = 16^{1/4}~[cos(\frac{180^{\circ}}{4}+\frac{(360^{\circ})(2)}{4})+i~sin(\frac{180^{\circ}}{4}+\frac{(360^{\circ})(2)}{4})]$
$z^{1/4} = 2~(cos~225^{\circ}+i~sin~225^{\circ})$
When k = 3:
$z^{1/4} = 16^{1/4}~[cos(\frac{180^{\circ}}{4}+\frac{(360^{\circ})(3)}{4})+i~sin(\frac{180^{\circ}}{4}+\frac{(360^{\circ})(3)}{4})]$
$z^{1/4} = 2~(cos~315^{\circ}+i~sin~315^{\circ})$
