If `|z_1|=2, |z_2|=3, |z_3|=4` and `|2z_1+3z_2 + 4z_3|=9` , then value of `|8z_2z_3 + 27z_3z_1 + 64z_1z_2|^(1//3)` is :

Answer

To solve the problem, we need to find the value of \( |8z_2z_3 + 27z_3z_1 + 64z_1z_2|^{1/3} \) given that \( |z_1| = 2 \), \( |z_2| = 3 \), \( |z_3| = 4 \), and \( |2z_1 + 3z_2 + 4z_3| = 9 \).

### Step-by-Step Solution:

1. **Identify the Moduli**:
We are given:
\[ |z_1| = 2, \quad |z_2| = 3, \quad |z_3| = 4 \]

2. **Express the Target Modulus**:
We need to find:
\[ |8z_2z_3 + 27z_3z_1 + 64z_1z_2|^{1/3} \]
We can factor out \( z_1 z_2 z_3 \):
\[ |z_1 z_2 z_3| \cdot |8/z_1 + 27/z_2 + 64/z_3| \]

3. **Calculate the Modulus of the Product**:
The modulus of the product is:
\[ |z_1 z_2 z_3| = |z_1| \cdot |z_2| \cdot |z_3| = 2 \cdot 3 \cdot 4 = 24 \]

4. **Calculate the Modulus of the Sum**:
We need to calculate:
\[ |8/z_1 + 27/z_2 + 64/z_3| \]
Using the property of modulus:
\[ |8/z_1| = \frac{8}{|z_1|} = \frac{8}{2} = 4, \quad |27/z_2| = \frac{27}{|z_2|} = \frac{27}{3} = 9, \quad |64/z_3| = \frac{64}{|z_3|} = \frac{64}{4} = 16 \]

5. **Combine the Terms**:
We can use the triangle inequality:
\[ |8/z_1 + 27/z_2 + 64/z_3| \leq |8/z_1| + |27/z_2| + |64/z_3| = 4 + 9 + 16 = 29 \]

6. **Use Given Information**:
We know from the problem that:
\[ |2z_1 + 3z_2 + 4z_3| = 9 \]
Therefore, we can express:
\[ |8/z_1 + 27/z_2 + 64/z_3| = |2z_1 + 3z_2 + 4z_3| = 9 \]

7. **Final Calculation**:
Now we can combine our results:
\[ |8z_2z_3 + 27z_3z_1 + 64z_1z_2| = |z_1 z_2 z_3| \cdot |8/z_1 + 27/z_2 + 64/z_3| = 24 \cdot 9 = 216 \]

8. **Take the Cube Root**:
Finally, we take the cube root:
\[ |8z_2z_3 + 27z_3z_1 + 64z_1z_2|^{1/3} = 216^{1/3} = 6 \]

### Final Answer:
\[ \boxed{6} \]