Evaluate `a^(x - y), a^(Y - z), a^(z - x)`

Answer

To evaluate the expression \( a^{(x - y)} \cdot a^{(y - z)} \cdot a^{(z - x)} \), we can follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ a^{(x - y)} \cdot a^{(y - z)} \cdot a^{(z - x)} \]

### Step 2: Use the property of exponents
Since the bases are the same (all are \( a \)), we can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \). Therefore, we can combine the exponents:
\[ a^{(x - y) + (y - z) + (z - x)} \]

### Step 3: Simplify the exponent
Now, we simplify the exponent:
\[ (x - y) + (y - z) + (z - x) \]

### Step 4: Combine like terms
When we combine the terms in the exponent, we can see that:
- The \( -y \) and \( y \) cancel each other out.
- The \( -z \) and \( z \) cancel each other out.
- The \( -x \) and \( x \) cancel each other out.

This gives us:
\[ 0 \]

### Step 5: Substitute back into the expression
Now we substitute back into the expression:
\[ a^0 \]

### Step 6: Evaluate \( a^0 \)
By the property of exponents, any non-zero number raised to the power of 0 is equal to 1:
\[ a^0 = 1 \]

### Final Answer
Thus, the value of \( a^{(x - y)} \cdot a^{(y - z)} \cdot a^{(z - x)} \) is:
\[ \boxed{1} \]