Let `a = 3 + 4i, z_(1) and z_(2)` be two complex numbers such that `|z_(1)| = 3 and |z_(2) - a| = 2`, then maximum possible value of `|z_(1) - z_(2)|` is ___________.

Answer

To solve the problem, we need to find the maximum possible value of \( |z_1 - z_2| \) given the conditions on the magnitudes of the complex numbers \( z_1 \) and \( z_2 \).

### Step-by-Step Solution:

1. **Understanding the given conditions**:
We have \( |z_1| = 3 \) and \( |z_2 - a| = 2 \) where \( a = 3 + 4i \).
This means \( z_1 \) lies on a circle of radius 3 centered at the origin (0,0), and \( z_2 \) lies on a circle of radius 2 centered at the point \( (3, 4) \).

2. **Visualizing the circles**:
- The circle for \( z_1 \) has a center at \( (0, 0) \) and a radius of 3.
- The circle for \( z_2 \) has a center at \( (3, 4) \) and a radius of 2.

3. **Finding the distance between the centers of the circles**:
The distance \( d \) between the centers of the two circles can be calculated using the distance formula:
\[ d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

4. **Calculating the maximum distance \( |z_1 - z_2| \)**:
The maximum distance \( |z_1 - z_2| \) occurs when \( z_1 \) and \( z_2 \) are positioned such that they are on the line connecting the centers of the circles, and they are at their farthest points from each other.
This maximum distance can be calculated as:
\[ |z_1 - z_2|_{\text{max}} = d + r_1 + r_2 \]
where \( r_1 \) is the radius of the circle for \( z_1 \) and \( r_2 \) is the radius of the circle for \( z_2 \).
Here, \( r_1 = 3 \) and \( r_2 = 2 \).

5. **Substituting the values**:
\[ |z_1 - z_2|_{\text{max}} = 5 + 3 + 2 = 10 \]

### Final Answer:
The maximum possible value of \( |z_1 - z_2| \) is **10**.