Answer
a) $\overline{z+w}=\overline{z}+\overline{w}$
b) $\overline{zw}=\overline{z} \cdot \overline{w}$
c) $\overline{z^n}=\overline{z}^{n}$
Work Step by Step
a) Consider $z=a+bi; w=c+di$
$\overline{z+w}=\overline{(a+bi)+(c+di)}$
or, $\overline{z+w}=(a+c)-(b+d)i$
or, $\overline{z+w}=(a-bi)+(c-di)$
Thus, $\overline{z+w}=\overline{z}+\overline{w}$
b) $z=a+bi; w=c+di$
$\overline{zw}=\overline{(a+bi)(c+di)}$
or, $\overline{zw}=ac-bd-bci-adi$
or, $\overline{zw}=(a-bi)(c-di)$
Thus, $\overline{zw}=\overline{z} \cdot \overline{w}$
c) Apply De Movire's Theorem
$\overline{z^n}=\overline{r^n(\cos n \theta+i \sin n \theta}$
or, $\overline{z^n}=[r(\cos \theta-i \sin \theta)]^n$
Thus, $\overline{z^n}=\overline{z}^{n}$
