Question

The roots of the equation $z^4+1=0$ are

• A. $(\pm 1\pm i)$
• B. $(\pm 2\pm 2i)$
• C. $\frac{1}{\sqrt{2}}(\pm 1\pm i)$
• D. None of these

Answer 1

Answer: (C)

We have, $z^4+1=0\Rightarrow z^4=-1$
$\Rightarrow z=(\cos \pi +i\sin \pi {)}^{1/4}$
$\Rightarrow z=\cos \frac{1}{4}(2k\pi +\pi )+i\sin \frac{1}{4}(2k\pi +\pi ),k=0,1,2,3$
$\Rightarrow z=\cos \frac{\pi }{4}+i\sin \frac{\pi }{4},\cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4}$
$\cos \frac{5\pi }{4}+i\sin \frac{5\pi }{4},\cos \frac{7\pi }{4}+i\sin \frac{7\pi }{4}$
$=\frac{1}{\sqrt{2}}(1+i),\frac{1}{\sqrt{2}}(-1+i),\frac{1}{\sqrt{2}}(-1-i),\frac{1}{\sqrt{2}}(1-i)$
Hence, the four roots of $z^4+1=0$ are $\frac{1}{\sqrt{2}}(\pm 1\pm i)$.