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
$(cos\theta +i\sin \theta {)}^n=cosn\theta +isinn\theta$
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]
$=cos\frac{\pi }{6}+isin\frac{\pi }{6}$
$=\frac{\sqrt{3}}{2}+\frac{i}{2}=\frac{\sqrt{3}+i}{2}$