For complex numbers `z_1 = 6+3i, z_2=3-I ` find `(z_1)/(z_2)`

Answer

To find the quotient of the complex numbers \( z_1 = 6 + 3i \) and \( z_2 = 3 - i \), we can follow these steps:

### Step 1: Write the division of complex numbers
We start with the expression:
\[ \frac{z_1}{z_2} = \frac{6 + 3i}{3 - i} \]

### Step 2: Multiply by the conjugate of the denominator
To simplify the division, we multiply both the numerator and the denominator by the conjugate of the denominator \( 3 + i \):
\[ \frac{6 + 3i}{3 - i} \cdot \frac{3 + i}{3 + i} = \frac{(6 + 3i)(3 + i)}{(3 - i)(3 + i)} \]

### Step 3: Expand the numerator
Now we will expand the numerator:
\[ (6 + 3i)(3 + i) = 6 \cdot 3 + 6 \cdot i + 3i \cdot 3 + 3i \cdot i \]
Calculating each term:
- \( 6 \cdot 3 = 18 \)
- \( 6 \cdot i = 6i \)
- \( 3i \cdot 3 = 9i \)
- \( 3i \cdot i = 3i^2 = 3(-1) = -3 \)

Combining these results:
\[ 18 + 6i + 9i - 3 = 15 + 15i \]

### Step 4: Expand the denominator
Now we expand the denominator:
\[ (3 - i)(3 + i) = 3^2 - i^2 = 9 - (-1) = 9 + 1 = 10 \]

### Step 5: Combine the results
Now we can combine the results from the numerator and denominator:
\[ \frac{15 + 15i}{10} \]

### Step 6: Simplify the expression
We can simplify this expression by dividing both the real and imaginary parts by 10:
\[ \frac{15}{10} + \frac{15i}{10} = \frac{3}{2} + \frac{3}{2}i \]

### Final Result
Thus, the result of \( \frac{z_1}{z_2} \) is:
\[ \frac{3}{2} + \frac{3}{2}i \]

---