If `x + 2y + 3z = 0 and x^(3) + 4y^(3) + 9z^(3)= 18 xyz`, evaluate: `((x+2y)^(2))/(xy) + ((2y + 3z)^(2))/(yz) + ((3z+ x)^(2))/(zx)`

Answer

To solve the problem, we need to evaluate the expression:

\[ \frac{(x + 2y)^2}{xy} + \frac{(2y + 3z)^2}{yz} + \frac{(3z + x)^2}{zx} \]

given the conditions:

1. \( x + 2y + 3z = 0 \)
2. \( x^3 + 4y^3 + 9z^3 = 18xyz \)

### Step 1: Express each term in the expression using the given condition

From the first condition \( x + 2y + 3z = 0 \), we can express each of the terms \( x + 2y \), \( 2y + 3z \), and \( 3z + x \):

- \( x + 2y = -3z \)
- \( 2y + 3z = -x \)
- \( 3z + x = -2y \)

### Step 2: Substitute these expressions into the original expression

Now substituting these into the expression:

\[ \frac{(-3z)^2}{xy} + \frac{(-x)^2}{yz} + \frac{(-2y)^2}{zx} \]

This simplifies to:

\[ \frac{9z^2}{xy} + \frac{x^2}{yz} + \frac{4y^2}{zx} \]

### Step 3: Combine the terms over a common denominator

The common denominator for these fractions is \( xyz \). Thus, we can rewrite the expression as:

\[ \frac{9z^2 \cdot z}{xyz} + \frac{x^2 \cdot x}{xyz} + \frac{4y^2 \cdot y}{xyz} \]

This gives us:

\[ \frac{9z^3 + x^3 + 4y^3}{xyz} \]

### Step 4: Substitute the second condition

From the second condition, we know that:

\[ x^3 + 4y^3 + 9z^3 = 18xyz \]

Thus, we can substitute this into our expression:

\[ \frac{18xyz}{xyz} \]

### Step 5: Simplify the expression

Now, simplifying gives:

\[ 18 \]

### Final Answer

Thus, the value of the expression is:

\[ \boxed{18} \]