If `(xy)^a=z,(yz)^a=x, and (xz)^a=y`, then what is the value of a? (None of x,y and z is either 0 or 1.)

Options

A. 1
B. `(1)/(2)`
C. `(3)/(2)`
D. 0

Explanation

To solve the problem, we need to find the value of \( a \) given the equations:

1. \((xy)^a = z\)
2. \((yz)^a = x\)
3. \((xz)^a = y\)

### Step 1: Write down the equations
We have:
- Equation 1: \((xy)^a = z\)
- Equation 2: \((yz)^a = x\)
- Equation 3: \((xz)^a = y\)

### Step 2: Multiply all three equations
We multiply all three equations together:

\[ (xy)^a \cdot (yz)^a \cdot (xz)^a = z \cdot x \cdot y \]

### Step 3: Simplify the left side
Using the property of exponents, we can combine the left side:

\[ (xy \cdot yz \cdot xz)^a = zxy \]

### Step 4: Expand the left side
Now, let's expand \( xy \cdot yz \cdot xz \):

\[ xy \cdot yz \cdot xz = x^2y^2z^2 \]

So we can rewrite the equation as:

\[ (x^2y^2z^2)^a = xyz \]

### Step 5: Rewrite the equation
This can be rewritten as:

\[ x^{2a}y^{2a}z^{2a} = xyz \]

### Step 6: Equate the powers
Since the bases are the same, we can equate the powers:

\[ 2a = 1 \quad \text{(for each of } x, y, z\text{)} \]

### Step 7: Solve for \( a \)
Now, solving for \( a \):

\[ a = \frac{1}{2} \]

### Conclusion
Thus, the value of \( a \) is \( \frac{1}{2} \).

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