The product of all the roots of {left( cos fr | vedclass

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(B) Let $z = \left( \cos \frac{\pi }{3} + i\sin \frac{\pi }{3} \right)^{3/4}$.Using De Moivre's Theorem,$z = \left( e^{i\pi/3} \right)^{3/4} = e^{i\pi

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$1$

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(B) Let $z = \left( \cos \frac{\pi }{3} + i\sin \frac{\pi }{3} ight)^{3/4}$.Using De Moivre's Theorem,$z = \left( e^{i\pi/3} ight)^{3/4} = e^{i\pi/4}$.The roots are given by $z_k = \cos \left( \frac{\pi + 2k\pi}{4} ight) + i\sin \left( \frac{\pi + 2k\pi}{4} ight)$ for $k = 0, 1, 2, 3$.These are the $4$th roots of $e^{i\pi} = -1$.The product of the $n$th roots of a complex number $w$ is given by $(-1)^{n-1} w$.Here,$n = 4$ and $w = -1$.Therefore,the product of the roots is $(-1)^{4-1} 	imes (-1) = (-1)^3 	imes (-1) = (-1) 	imes (-1) = 1$.

The product of all the roots of ${\left( \cos \frac{\pi }{3} + i\sin \frac{\pi }{3} \right)^{3/4}}$ is

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