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 where \( a = b^x \), \( b = c^y \), and \( c = a^z \), we will express all variables in terms of one variable and find the value of \( xyz \).

### Step 1: Substitute \( b \) and \( c \) in terms of \( a \)

Given:
1. \( a = b^x \)
2. \( b = c^y \)
3. \( c = a^z \)

We can substitute \( b \) and \( c \) into the first equation.

From equation 2, substitute \( b \):
\[ b = c^y = (a^z)^y = a^{zy} \]
Now substitute \( b \) back into equation 1:
\[ a = (a^{zy})^x = a^{zyx} \]

### Step 2: Equate the exponents

Since \( a \) is not zero, we can equate the exponents:
\[ 1 = zyx \]

### Step 3: Solve for \( xyz \)

From the equation \( zyx = 1 \), we can rearrange it to find \( xyz \):
\[ xyz = 1 \]

### Conclusion

Thus, the value of \( xyz \) is \( 1 \).

### Final Answer
The answer is \( xyz = 1 \).