Question

z = ( root3/2 + i/2 )^5 + ( root3/2 - i/2 )^5 then (a) Rez = 0 (b) Imz = 0 (c) Rez>0 Imz>0 (d) Rez>0 Imz<0 I WILL MARK U AS BRAINLIEST PLZ ANSWER CORRECTLY​

Answer 1

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Answer 2

Answer:

(b) Im(z) = 0

Step-by-step explanation:

$z=(\frac{\sqrt3}{2}+i\frac{1}{2})^5 +(\frac{\sqrt3}{2}-i\frac{1}{2})^5\\\\=(\cos {\frac{\pi}{6}} + i \sin {\frac{\pi}{6}})^5+(\cos {\frac{-\pi}{6}} + i \sin {\frac{-\pi}{6}})^5\\=(e^{i\frac{\pi}{6}})^5+(e^{-i\frac{\pi}{6}})^5$

$=e^{i\frac{5\pi}{6}}+e^{-i\frac{5\pi}{6}}$

$=\cos {\frac{5\pi}{6}}+i\sin{\frac{5\pi}{6}}+\cos{\frac{-5\pi}{6}}+i\sin{\frac{-5\pi}{6}}\\=-\frac{\sqrt3}{2}+i\frac{1}{2}-\frac{\sqrt3}{2}-i\frac{1}{2}\\=-\sqrt3$

Thus we find that: $z=-\sqrt3$

This is a real no. so clearly:  Im(z)=0

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