If `a = b^x , b = c^y, c = a^z`, then xyz is :

Options

A. `-1`
B. `0`
C. `1`
D. `abc`

Answer

To solve the problem, we need to express the relationships given in the question mathematically and derive the value of \(xyz\).

### Step-by-Step Solution:

1. **Write down the given equations:**
\[ a = b^x \quad (1) \]
\[ b = c^y \quad (2) \]
\[ c = a^z \quad (3) \]

2. **Substitute equation (2) into equation (1):**
From equation (2), we can express \(b\) in terms of \(c\):
\[ b = c^y \]
Substitute this into equation (1):
\[ a = (c^y)^x = c^{xy} \quad (4) \]

3. **Substitute equation (3) into equation (4):**
From equation (3), we can express \(c\) in terms of \(a\):
\[ c = a^z \]
Substitute this into equation (4):
\[ a = (a^z)^{xy} = a^{xyz} \quad (5) \]

4. **Equate the powers of \(a\):**
Since we have \(a = a^{xyz}\), we can equate the exponents:
\[ 1 = xyz \quad (6) \]

5. **Conclusion:**
Therefore, we find that:
\[ xyz = 1 \]

### Final Answer:
The value of \(xyz\) is \(1\).

---