Question

Find all the values of $(1+i\surd 3{)}^{1/5}$.

Answer 1

Given, $z^5=(1+i=\sqrt{3})$

$=2|\frac{1}{2}+i\frac{\sqrt{3}}{2}|=2[\cos \left(\frac{\pi }{3}\right)+i\sin \left(\frac{\pi }{3}\right)]$

Here, $r=2,\varphi =arg\left(z\right)=\pi /3,n=5$

$n^{th}$ roots are given by $r^{1/5}[\cos \left(\frac{\varphi +2k\pi }{n}\right)+i\sin \left(\frac{\varphi +2k\pi }{n}\right)]$

$z_0=2^{1/5}\left[\frac{\cos \frac{\pi }{3}+2\left(0\right)\pi }{5}\right]+i\left[\frac{\sin \frac{\pi }{3}+2\left(0\right)\pi }{5}\right]$

$z_0=2^{1/5}(\cos \frac{\pi }{15}+i\sin \frac{\pi }{15})$

$z_1=2^{1/5}[\frac{\cos \frac{\pi }{3}+2\pi }{5}+\frac{i\sin \frac{\pi }{3}+2\pi }{5}]=2^{1/5}|\cos \frac{7\pi }{15}+i\sin \frac{7\pi }{15}]$

$z_2=2^{1/5}[\frac{\cos \frac{\pi }{3}+4\pi }{5}+\frac{i\sin \frac{\pi }{3}+4\pi }{5}]=2^{1/5}|\cos \frac{13\pi }{15}+i\sin \frac{3\pi }{15}]$

$z_3=2^{1/5}[\cos \frac{19\pi }{15}+i\sin \frac{19\pi }{15}]$ [ putting $k=3$ ]

$z_4=2^{1/5}[\cos \frac{25\pi }{15}+i\sin \frac{25\pi }{15}]$ [ putting $k=4$ ]

$z_4=2^{1/5}|\cos \frac{5\pi }{3}+i\sin \frac{5\pi }{3}|$

$z_0,z_1,z_2,z_3,z_4$ are the fifth roots of given complex no.