The value of `((a^(x))/(a^(y)))^(x^(2) + xy +y^(2)) xx ((a^(y))/(a^(z)))^(y^(2) + yz + z^(2)) xx ((a^(z))/(a^(x)))^(z^(2) + xz + x^(2))` is

Options

A. `a^(xy)`
B. 1
C. `a^(x)`
D. 0

Answer

To solve the given expression

\[ \left(\frac{a^x}{a^y}\right)^{x^2 + xy + y^2} \times \left(\frac{a^y}{a^z}\right)^{y^2 + yz + z^2} \times \left(\frac{a^z}{a^x}\right)^{z^2 + xz + x^2} \]

we will follow these steps:

### Step 1: Simplify Each Fraction
Using the property of indices, we can simplify each fraction:

\[ \frac{a^x}{a^y} = a^{x-y}, \quad \frac{a^y}{a^z} = a^{y-z}, \quad \frac{a^z}{a^x} = a^{z-x} \]

### Step 2: Rewrite the Expression
Now we can rewrite the entire expression:

\[ \left(a^{x-y}\right)^{x^2 + xy + y^2} \times \left(a^{y-z}\right)^{y^2 + yz + z^2} \times \left(a^{z-x}\right)^{z^2 + xz + x^2} \]

### Step 3: Apply the Power of a Power Rule
Using the property \((a^m)^n = a^{mn}\), we can further simplify:

\[ a^{(x-y)(x^2 + xy + y^2)} \times a^{(y-z)(y^2 + yz + z^2)} \times a^{(z-x)(z^2 + xz + x^2)} \]

### Step 4: Combine the Exponents
Now we can combine the exponents:

\[ a^{(x-y)(x^2 + xy + y^2) + (y-z)(y^2 + yz + z^2) + (z-x)(z^2 + xz + x^2)} \]

### Step 5: Analyze the Exponent
We need to analyze the expression inside the exponent:

Let:
- \(A = (x-y)(x^2 + xy + y^2)\)
- \(B = (y-z)(y^2 + yz + z^2)\)
- \(C = (z-x)(z^2 + xz + x^2)\)

We will show that \(A + B + C = 0\).

### Step 6: Expand Each Term
Expanding \(A\):

\[ A = (x-y)(x^2 + xy + y^2) = x^3 - y^3 \]

Similarly, we can expand \(B\) and \(C\):

\[ B = (y-z)(y^2 + yz + z^2) = y^3 - z^3 \]
\[ C = (z-x)(z^2 + xz + x^2) = z^3 - x^3 \]

### Step 7: Combine the Results
Now we combine \(A\), \(B\), and \(C\):

\[ A + B + C = (x^3 - y^3) + (y^3 - z^3) + (z^3 - x^3) = 0 \]

### Step 8: Final Result
Thus, the exponent simplifies to \(0\):

\[ a^0 = 1 \]

Therefore, the value of the entire expression is:

\[ \boxed{1} \]