If a, b, and c are not equal to 0 or 1 and if `a^x = b, b^y = c and c^x = a` then `xyz` =

Options

A. 0
B. 1
C. 2
D. a

Answer

To solve the problem, we start with the given equations:

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

We need to find the value of \( xyz \).

### Step 1: Substitute \( b \) into the second equation
From the first equation, we have \( b = a^x \). We can substitute this value into the second equation:

\[ b^y = c \implies (a^x)^y = c \]

### Step 2: Simplify the expression
Using the property of exponents \( (x^m)^n = x^{m \cdot n} \), we can rewrite the equation:

\[ a^{xy} = c \]

### Step 3: Substitute \( c \) into the third equation
Now we have \( c = a^{xy} \). We substitute this into the third equation:

\[ c^z = a \implies (a^{xy})^z = a \]

### Step 4: Simplify the expression again
Again, using the property of exponents, we can rewrite the equation:

\[ a^{xyz} = a^1 \]

### Step 5: Compare the exponents
Since the bases are the same (both are \( a \)), we can equate the exponents:

\[ xyz = 1 \]

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

### Final Answer
\[ xyz = 1 \]

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