Find the complex number, lying in the second quadrant, and having the smallest possible real part, which satisfies the equation
$w^8=15-15i$
Find the complex number, lying in the second quadrant, and having the smallest possible real part, which satisfies the equation
$w^8=15-15i$
$z=15-15i\Longrightarrow |z|=15\,\sqrt 2\,\exp({7\pi i}/{4}+2k\pi i),\,k\in\Bbb Z$
$\Longrightarrow w^8=z\Longrightarrow w=z^{1/8}=15^{1/8}\,2^{1/16}\,\exp({7\pi i}/32+{k\pi i}/{4})$
Now just observe that as $\,k\,$ runs from $\,0\,$ to $\,7\,$, we get all the possible (eight) values on the right-hand side above...
Hint: use the polar form of complex numbers.