If `|z_1|=1, |z_2|=2, |z_3|=3` and `|9z_1z_2 + 4z_1z_3 + z_2z_3|=36`, then `|z_1+z_2+z_3|` is equal to :

Answer

To solve the problem step by step, we will use the properties of complex numbers and their moduli.

### Given:
- \( |z_1| = 1 \)
- \( |z_2| = 2 \)
- \( |z_3| = 3 \)
- \( |9z_1z_2 + 4z_1z_3 + z_2z_3| = 36 \)

### Step 1: Rewrite the modulus equation
We start with the equation:
\[ |9z_1z_2 + 4z_1z_3 + z_2z_3| = 36 \]

### Step 2: Substitute the moduli
Using the properties of moduli, we can express the terms:
- \( |z_1|^2 = 1^2 = 1 \)
- \( |z_2|^2 = 2^2 = 4 \)
- \( |z_3|^2 = 3^2 = 9 \)

Now, rewrite the equation:
\[ |9|z_1||z_2| + 4|z_1||z_3| + |z_2||z_3| = |9 \cdot 1 \cdot 2 + 4 \cdot 1 \cdot 3 + 2 \cdot 3| \]
This simplifies to:
\[ |18 + 12 + 6| = |36| \]

### Step 3: Factor out the moduli
Now we can factor out the moduli:
\[ |z_1 z_2 z_3| = |z_1| \cdot |z_2| \cdot |z_3| = 1 \cdot 2 \cdot 3 = 6 \]

### Step 4: Use the property of moduli
We know that:
\[ |z_1 z_2 z_3| \cdot |z_1^* + z_2^* + z_3^*| = 36 \]
Where \( z_i^* \) denotes the conjugate of \( z_i \).

### Step 5: Solve for the sum of moduli
Substituting the known modulus:
\[ 6 \cdot |z_1^* + z_2^* + z_3^*| = 36 \]
Dividing both sides by 6 gives:
\[ |z_1^* + z_2^* + z_3^*| = 6 \]

### Step 6: Final result
Since \( |z_1 + z_2 + z_3| = |z_1^* + z_2^* + z_3^*| \), we conclude that:
\[ |z_1 + z_2 + z_3| = 6 \]

### Final Answer:
\[ |z_1 + z_2 + z_3| = 6 \]

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