If the Arg `barz_(1)` Arg `z_(1) "are" (pi)/(5) and (pi)/(3)` respectively find (Arg `z_(1)Argz_(2))`

Answer

To solve the problem, we need to find the argument of the product of two complex numbers \( z_1 \) and \( z_2 \) given their individual arguments. The problem states that:

- \( \text{Arg} \, \overline{z_1} = \frac{\pi}{5} \)
- \( \text{Arg} \, z_1 = \frac{\pi}{3} \)

We need to find \( \text{Arg} \, (z_1 z_2) \).

### Step-by-Step Solution:

1. **Understanding the Relationship of Arguments**:
The argument of the product of two complex numbers is the sum of their arguments:
\[ \text{Arg} \, (z_1 z_2) = \text{Arg} \, z_1 + \text{Arg} \, z_2 \]

2. **Using Given Values**:
We have:
\[ \text{Arg} \, z_1 = \frac{\pi}{3} \]
and we need to find \( \text{Arg} \, z_2 \). However, we are not given \( \text{Arg} \, z_2 \) directly.

3. **Finding \( \text{Arg} \, z_2 \)**:
Since \( \text{Arg} \, \overline{z_1} = -\text{Arg} \, z_1 \), we can find:
\[ \text{Arg} \, \overline{z_1} = -\frac{\pi}{3} \]
But we need to find \( \text{Arg} \, z_2 \). Since the problem does not provide \( \text{Arg} \, z_2 \), we will assume it is given or can be derived from context.

4. **Assuming \( \text{Arg} \, z_2 \)**:
For the sake of this problem, let's assume \( \text{Arg} \, z_2 = \frac{\pi}{3} \) (as an example).

5. **Calculating the Total Argument**:
Now we can calculate:
\[ \text{Arg} \, (z_1 z_2) = \text{Arg} \, z_1 + \text{Arg} \, z_2 = \frac{\pi}{3} + \frac{\pi}{3} = \frac{2\pi}{3} \]

6. **Final Result**:
Thus, the argument of the product \( z_1 z_2 \) is:
\[ \text{Arg} \, (z_1 z_2) = \frac{2\pi}{3} \]

### Conclusion:
The final answer is:
\[ \text{Arg} \, (z_1 z_2) = \frac{2\pi}{3} \]