Prove that the following are real numbers 
  
*
  
*$$\frac{1}{z}+\frac{1}{\overline{z}}$$
  
*$$z^3\cdot\overline{z}+z\cdot\overline{z}^3$$
  

1.$$\frac{1}{z}+\frac{1}{\overline{z}}$$
$$\frac{\overline{z}}{z\cdot \overline{z}}+\frac{z}{z\cdot\overline{z}}$$
$$\frac{\overline{z}+z}{z\cdot \overline{z}}$$
$$\frac{2Re(z)}{|z|}$$ which is real number as a division of two real numbers


*$$z^3\cdot\overline{z}+z\cdot\overline{z}^3$$


$$\overline{z^3\cdot\overline{z}+z\cdot\overline{z}^3}$$
Due to: $\overline{z_{1}+z_{2}}=\overline{z_{1}}+\overline{z_{2}}$ and $\overline{z_{1}\cdot z_{2}}=\overline{z_{1}}\cdot\overline{z_{2}}$
we get:
$$\overline{z}^3\cdot{z}+\overline{z}\cdot{z}^3$$
Now we can say that we got the same elements and $(z=\overline{z})$ therefore it is real number? 
Is the reasoning in both of these correct?

If we know $z + \overline z \in \mathbb R$ and $z*\overline z = |z|^2\in \mathbb R$ and $\overline{\overline z} = z$ and $\overline {z*w} = \overline z*\overline w$   we are pretty much done.
1) $\frac 1{\overline z} *\overline z = 1$
$\overline{\frac 1{\overline z}*\overline z} = \overline 1 = 1$
$\overline{\frac 1{\overline z}}*\overline{\overline z} = \overline{\frac 1{\overline z}}*z = 1$
$\overline{\frac 1{\overline z}} = \frac 1z$ so 
$\frac {1}{\overline z} = \overline {\frac 1z}$ so
$\frac 1z + \frac 1{\overline z} = \frac 1z + \overline {\frac 1z} \in \mathbb R$.
2) $z^3*\overline z +z*\overline{z^3}= \overline{\overline {z^3}}*\overline z +\overline{z^3}*z= \overline{\overline {z^3}*z}+\overline{z^3}*z\in \mathbb R$
