Question

Find the number of roots of the equation $z^6+64=0$ where real part is positive.

Answer 1

$z^6=-64$

$z^6=2^6.e^{i(2n+1)\pi }$ $n=0,1,2,3,4,5$

$\Rightarrow$ $z=2e^{i(2n+1)\frac{\pi }{6}}$

$\therefore$ $z=2e^{i\frac{\pi }{6}},2e^{i\frac{\pi }{2}},2e^{i\frac{5\pi }{6}},2e^{i\frac{7\pi }{6}},2e^{i\frac{3\pi }{2}},2e^{i\frac{11\pi }{6}}$

$\therefore$ roots with +ve real part are$=2e^{\frac{i\pi }{6}},2e^{i\frac{11\pi }{6}}$