Question

what is the value of b​

Answer 1

Answer:

$( \sqrt{3} + i)^{100} = 2^{99} (a + ib) \\ (2( \frac{ \sqrt{3} }{2} + \frac{i}{2} ))^{100} = 2^{99} (a + ib) \\ ( \cos( \frac{\pi}{6} ) + i \: sin( \frac{\pi}{6} ))^{100} = ( \frac{a}{2} + \frac{ib}{2} ) \\ \cos( \frac{100\pi}{6} ) + i \: sin( \frac{100\pi}{6} ) = ( \frac{a}{2} + \frac{ib}{2} ) \\ equating \: imazinary \: part \\ \frac{b}{2} = \sin( \frac{100\pi}{6} ) \\ b = \sqrt{3} \: \: (ans)$