For any two complex numbers $z_{1}$ and $z_{2},$ prove that

$\operatorname{Re}\left(z_{1} z_{2}\right)=\operatorname{Re} z_{1} \operatorname{Re} z_{2}-\operatorname{Im} z_{1} \operatorname{Im} z_{2}$

Let $S_{1}=\left\{z_{1} \in C:\left|z_{1}-3\right|=\frac{1}{2}\right\}$ and $S_{2}=\left\{z_{2} \in C:\left|z_{2}-\right| z_{2}+1||=\left|z_{2}+\right| z_{2}-1||\right\} . \quad$ Then, for $z_{1} \in S_{1}$ and $z_{2} \in S_{2}$, the least value of $\left|z_{2}-z_{1}\right|$ is.

For positive integers ${n_1},{n_2}$the value of the expression ${(1 + i)^{{n_1}}} + {(1 + {i^3})^{{n_1}}} + {(1 + {i^5})^{{n_2}}} + {(1 + {i^7})^{{n_2}}}$where $i = \sqrt { – 1} $ is a real number if and only if

Solve the equation $3 x^{2}-4 x+\frac{20}{3}=0$

If ${z_1},{z_2} \in C,$ then