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Let z and w be two non- zero complex numbers such that |z|=|w| and arg(z)+arg(w)=π. Then the value of (z+¯¯¯¯w)10 is

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Solution

Let arg(w)=θ. Then arg(z)=πθ.
w=|w|(cosθ+isinθ)
and
z=|z|(cos(πθ)+isin(πθ))
=|w|(- cosθ+i sinθ)[|z|=|w|]
=|w|(cosθi sinθ)=¯¯¯¯wz+¯¯¯¯w=0.
(z+¯¯¯¯w)10=0

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