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
Let us consider that $z=a+bi; w=c+di$
a) $\overline{z+w}=\overline{(a+bi)+(c+di)}$
Re-write as: $\overline{z+w}=(a-bi)+(c-di)$
Thus, we get $\overline{z+w}=\overline{z}+\overline{w}$
b) $\overline{zw}=\overline{(a+bi)(c+di)}$
Re-write as: $\overline{zw}=ac-bd-bci-adi$
Thus, we get $\overline{zw}=\overline{z} \cdot \overline{w}$
c) We need to use De Movire's Theorem.
we have $\overline{z^n}=\overline{r^n(\cos n \theta+i \sin n \theta}$
This implies that $\overline{z^n}=[r(\cos \theta-i \sin \theta)]^n$
Thus, we get $\overline{z^n}=\overline{z}^{n}$
