What is the imaginary part of \(\overline{(2+i)^{-1}}\)?
Express your answer
as a decimal.
The answer is \(0.2\). To see this,
note that
\begin{eqnarray}
\displaystyle(2+i)^{-1}
& = & \frac{1}{2+i} \\
& = & \frac{1}{2+i}\cdot\frac{\overline{2+i}}{\overline{2+i}} \\
& = & \frac{1}{2+i}\cdot\frac{2-i}{2-i} \\
& = & \frac{2-i}{5} \\
& = & 0.4-0.2i.\end{eqnarray}
So \(\overline{(2+i)^{-1}}=\overline{0.4-0.2i}=0.4+0.2i\).