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Can someone help me in approaching and solving this math problem?

For any integer \(n\) the argument of \[z=\dfrac{(\sqrt{3}+i)^{4n+1}}{(1-i\sqrt{3})^{4n}}\] is:- \[(a)\dfrac{\pi}{6}\]

\[(b)\dfrac{\pi}{3}\]

\[(c)\dfrac{\pi}{2}\]

\[(d)\dfrac{2\pi}{3}\]

This question is based on COMPLEX NUMBERS..

Please post the solution in Detail...

Note by Parag Zode
3 years ago

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\(z=\dfrac{(2 cis (\frac{\pi}{6}))^{4n+1}}{(2 cis (-\frac{\pi}{3}))^{4n}}\)

\(z=\left(\dfrac{2 cis (\frac{\pi}{6})}{2 cis (-\frac{\pi}{3})}\right)^{4n} 2 cis (\frac{\pi}{6})\)

\(z=2(cis (\frac{\pi}{2}))^{4n} cis (\frac{\pi}{6})\)

\(z=2(i)^{4n} cis (\frac{\pi}{6})\)

\(z=2 cis (\frac{\pi}{6})\)

\(\arg z = \boxed{\dfrac{\pi}{6}}\)

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Is the Answer Pi/6 ???

Abhineet Nayyar - 3 years ago

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