Question

The value of $i^{1/3}$ is:

• A. $\frac{\sqrt{3}+i}{2}$
• B. $\frac{\sqrt{3}-i}{2}$
• C. $\frac{1+i \sqrt{3}}{2}$
• D. $\frac{1-i \sqrt{3}}{2}$

Answer 1

Answer: (A)

De' Moiver's theorem : If $n$ is any rational number, then
$(cos\, \theta+i \,\sin \,\theta)^{n}=cos\, n \,\theta+i \,sin \,n \,\theta$
We have, $i^{1 / 3}=\left(cos \frac{\pi}{2}+i \,\sin \frac{\pi}{2}\right)^{1 / 3}$
$=cos \frac{1}{3} \frac{\pi}{2}+i \sin \frac{1}{3} \frac{\pi}{2}$
[By using above given theorem]
$=cos \frac{\pi}{6}+i \,sin \frac{\pi}{6}$
$=\frac{\sqrt{3}}{2}+\frac{i}{2}=\frac{\sqrt{3}+i}{2}$