Answer
$\displaystyle \frac{7}{4}+\frac{1}{4}i$
Work Step by Step
$\displaystyle \frac{1}{Z_{1}}=\frac{1}{2+i}\cdot\frac{2-i}{2-i}=\frac{2-i}{2^{2}+1^{2}}=\frac{2}{5}-\frac{1}{5}i$
$\displaystyle \frac{1}{Z_{2}}=\frac{1}{4-3i}\cdot\frac{4+3i}{4+3i}=\frac{4+3i}{4^{2}+3^{2}} =\frac{4}{25}+\frac{3}{25}i$
$\displaystyle \frac{1}{Z_{1}}+\frac{1}{Z_{2}}=(\frac{2}{5}+\frac{4}{25})+(-\frac{1}{5}+\frac{3}{25})i=\frac{14}{25}-\frac{2}{25}i$
So
$\displaystyle \frac{1}{Z}=\frac{14-2i}{25}\Rightarrow Z=\frac{25}{14-2i}$
$Z=\displaystyle \frac{25}{14-2i}\cdot\frac{14+2i}{14+2i}=\frac{25(14+2i)}{14^{2}+2^{2}}=\frac{350+50i}{200}$
$Z=\displaystyle \frac{350}{200}+\frac{50i}{200}=\frac{7}{4}+\frac{1}{4}i$
