Question

Find all those roots of the equation $z^{12}-56z^6-512=0$ whose imaginary part is positive.

• A. $2,2(cos\frac{\pi }{3}+isin\frac{\pi }{3}),2(cos\frac{2\pi }{3}+isin\frac{2\pi }{3}),-2,$$2^{2/3}(cos\frac{\pi }{6}+isin\frac{\pi }{6}),2^{2/3}(cos\frac{\pi }{2}+isin\frac{\pi }{2}),2^{2/3}(cos\frac{5\pi }{6}+isin\frac{5\pi }{6})$
• B. $2,2(cos\frac{\pi }{3}+isin\frac{\pi }{3}),2(cos\frac{2\pi }{3}+isin\frac{2\pi }{3}),-2,$$2^{1/3}(cos\frac{\pi }{6}+isin\frac{\pi }{6}),2^{1/3}(cos\frac{\pi }{2}+isin\frac{\pi }{2}),2^{1/3}(cos\frac{5\pi }{6}+isin\frac{5\pi }{6})$
• C. $2,2(cos\frac{\pi }{3}+isin\frac{\pi }{3}),2(cos\frac{2\pi }{3}+isin\frac{2\pi }{3}),-2,$$2^{1/3}(-cos\frac{\pi }{6}+isin\frac{\pi }{6}),2^{1/3}(-cos\frac{\pi }{2}+isin\frac{\pi }{2}),2^{1/3}(-cos\frac{5\pi }{6}+isin\frac{5\pi }{6})$
• D. $2,2(-cos\frac{\pi }{3}+isin\frac{\pi }{3}),2(-cos\frac{2\pi }{3}+isin\frac{2\pi }{3}),-2,$$2^{2/3}(cos\frac{\pi }{6}+isin\frac{\pi }{6}),2^{2/3}(cos\frac{\pi }{2}+isin\frac{\pi }{2}),2^{2/3}(cos\frac{5\pi }{6}+isin\frac{5\pi }{6})$

Answer 1

The correct option is B $2,2(cos\frac{\pi }{3}+isin\frac{\pi }{3}),2(cos\frac{2\pi }{3}+isin\frac{2\pi }{3}),-2,$

$2^{1/3}(cos\frac{\pi }{6}+isin\frac{\pi }{6}),2^{1/3}(cos\frac{\pi }{2}+isin\frac{\pi }{2}),2^{1/3}(cos\frac{5\pi }{6}+isin\frac{5\pi }{6})$

$(z^6-28{)}^2-784-512=0$
$(z^6-28{)}^2=1296$
$z^6-28=\pm 36$
$z^6=64$ and $z^6=-8$
$z^3=\pm 8$
$z=2$ and $z=-2$ ...(i)
$z^6=2^3.e^{i(2k-1)\pi }$
$z=2^{\frac{1}{2}}\left(e^{i\frac{(2k-1)\pi }{6}}\right)$ where $k=1,2,\mathrm{3..6}$.