Question

If $x^6=(4-3i{)}^5$, then the product of all of its roots is (where $\theta =-{\tan }^{-1}(3/4))$

• A. $5^5(\cos 5\theta +i\sin 5\theta )$
• B. $-5^5(\cos 5\theta +i\sin 5\theta )$
• C. $5^5(\cos 5\theta -i\sin 5\theta )$
• D. $-5^5(\cos 5\theta -i\sin 5\theta )$

Answer 1

Answer: (B)

$x^6=(4-3i{)}^5$
$x^6=5^6\left(\frac{4}{5}-\frac{3i}{5}\right)=5^5(\cos \theta +i\sin \theta {)}^5$
where $\theta =-{\tan }^{-1}\left(\frac{3}{4}\right)=5^5(\cos 5\theta +i\sin 5\theta )$
$x=5^{5/6}(\cos 5\theta +i\sin 5\theta {)}^{1/6}$
$=5^{5/6}\left[\cos \left(\frac{2k\pi +5\theta }{6}\right)+i\sin \left(\frac{2k\pi +5\theta }{6}\right)\right]$
$x_1{x}_2\dots x_6=5^5[\cos (5\pi +5\theta )+i\sin (5\pi +5\theta )]$
$=5^5(-\cos 5\theta -i\sin 5\theta )$
$=-5^5(\cos 5\theta +i\sin 5\theta )$