Answer
$[2{cis45^{\circ},2cis 135^{\circ},2cis 225^{\circ},2cis 315^{\circ}}]$
Work Step by Step
Given: $x^{4}+16=0$
or $x^{4}=-16$
$-16$ can be written in trigonometric form as:
$-16=16(\cos 180^{\circ}+\sin 180^{\circ})$
$x^{4}=16(\cos 180^{\circ}+\sin 180^{\circ})$
$x^{4}=2^{4}[\cos (180^{\circ}+2k\pi)+i \sin (180^{\circ}+2k\pi)]$
$x=2^{4\times \frac{1}{4}}[(\cos (180^{\circ}+2k\pi)+i \sin (180^{\circ}+2k\pi))]^{\frac{1}{4}}$
$x=2[(\cos (180^{\circ}+2k\pi)+i \sin (180^{\circ}+2k\pi))]^{\frac{1}{4}}$
Apply De-Moivre's Theorem
$x=2[(\cos \frac{(180^{\circ}+2k\pi)}{4}+i \sin\frac{(180^{\circ}+2k\pi)}{4})]$
Now, the arguments can be written as: $\frac{180^{\circ}+2k\pi}{4} $ for $k=0,1,2,3$
Arguments are:
$45^{\circ}$ for $k=0$
$135^{\circ}$ for $k=1$
$225^{\circ}$ for $k=2$
$315^{\circ}$ for $k=3$
Solution set of the equation can be written as:
$[2{cis45^{\circ},2cis 135^{\circ},2cis 225^{\circ},2cis 315^{\circ}}]$
