Question

If $i z^{4}+1=0$, then $z$ can take the value

• A. $\frac{1+i}{\sqrt{2}}$
• B. $\cos \frac{\pi}{8}+I \sin \frac{\pi}{8}$
• C. $\frac{1}{4 i}$
• D. $i$

Answer 1

Answer: (B)

$i z^{4}=-1$
$\Rightarrow z^{4}=-\frac{1}{i}$
$\Rightarrow z^{4}=i$
$\Rightarrow z=(i)^{1 / 4}=(0+i)^{1 / 4}$
$\Rightarrow z=\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)^{1 / 4}$
$\Rightarrow z=\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}$
(using De Moirrels theorem)