If `a^(x)=b^(y)=c^(z)` and abc = 1, then what is the value of xy + yz + zx ?

Options

A. 1
B. 3
C. 0
D. 5

Answer

To solve the problem given that \( a^x = b^y = c^z \) and \( abc = 1 \), we can follow these steps:

### Step 1: Set a common value
Let \( a^x = b^y = c^z = k \). This means we can express \( a \), \( b \), and \( c \) in terms of \( k \):
- \( a = k^{1/x} \)
- \( b = k^{1/y} \)
- \( c = k^{1/z} \)

### Step 2: Use the condition \( abc = 1 \)
Substituting the expressions for \( a \), \( b \), and \( c \) into the equation \( abc = 1 \):
\[ (k^{1/x})(k^{1/y})(k^{1/z}) = 1 \]
This simplifies to:
\[ k^{(1/x) + (1/y) + (1/z)} = 1 \]

### Step 3: Analyze the equation
For \( k^{(1/x) + (1/y) + (1/z)} = 1 \) to hold true, the exponent must be zero:
\[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 0 \]

### Step 4: Find a common denominator
To combine the fractions, we can find a common denominator, which is \( xyz \):
\[ \frac{yz + zx + xy}{xyz} = 0 \]
This implies that the numerator must be zero:
\[ yz + zx + xy = 0 \]

### Step 5: Conclusion
Thus, we have found that:
\[ xy + yz + zx = 0 \]

### Final Answer
The value of \( xy + yz + zx \) is \( \boxed{0} \).

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