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

Options

A. 3
B. 0
C. 1
D. can't be determined

Answer

To solve the problem, we need to analyze the given equations step by step.

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

### Step 1: Express \( b \) in terms of \( a \)
From the first equation, we can express \( b \) as:
\[ b = a^x \]

**Hint:** Substitute the expression for \( b \) into the next equation.

### Step 2: Substitute \( b \) into the second equation
Now substitute \( b = a^x \) into the second equation:
\[ (a^x)^y = c \]
Using the property of exponents \((m^p)^q = m^{p \cdot q}\), we get:
\[ a^{xy} = c \]

**Hint:** Now express \( c \) in terms of \( a \).

### Step 3: Express \( c \) in terms of \( a \)
From the second equation, we have:
\[ c = a^{xy} \]

**Hint:** Substitute \( c \) into the third equation.

### Step 4: Substitute \( c \) into the third equation
Now substitute \( c = a^{xy} \) into the third equation:
\[ (a^{xy})^z = a \]
Again, using the property of exponents:
\[ a^{xyz} = a^1 \]

**Hint:** Since the bases are the same, you can equate the exponents.

### Step 5: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[ xyz = 1 \]

### Conclusion
Thus, the value of \( xyz \) is:
\[ \boxed{1} \]