Expression `((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+zx+x^2)` in its simplest force is

Options

A. 1. a
B. 2. `1/a`
C. 3. 0
D. 4. 1

Answer

To simplify the expression \(\left(\frac{a^x}{a^y}\right)^{x^2 + xy + y^2} \times \left(\frac{a^y}{a^z}\right)^{y^2 + yz + z^2} \times \left(\frac{a^z}{a^x}\right)^{z^2 + zx + x^2}\), we can follow these steps:

### Step 1: Simplify Each Fraction
We start by simplifying each fraction in the expression:
\[ \frac{a^x}{a^y} = a^{x-y}, \quad \frac{a^y}{a^z} = a^{y-z}, \quad \frac{a^z}{a^x} = a^{z-x} \]

### Step 2: Substitute Back into the Expression
Now we substitute these simplified fractions back into the original expression:
\[ \left(a^{x-y}\right)^{x^2 + xy + y^2} \times \left(a^{y-z}\right)^{y^2 + yz + z^2} \times \left(a^{z-x}\right)^{z^2 + zx + x^2} \]

### Step 3: Apply the Power of a Power Rule
Using the power of a power rule, we can simplify further:
\[ a^{(x-y)(x^2 + xy + y^2)} \times a^{(y-z)(y^2 + yz + z^2)} \times a^{(z-x)(z^2 + zx + x^2)} \]

### Step 4: Combine the Exponents
Since the bases are the same, we can combine the exponents:
\[ a^{(x-y)(x^2 + xy + y^2) + (y-z)(y^2 + yz + z^2) + (z-x)(z^2 + zx + x^2)} \]

### Step 5: Recognize the Pattern
Notice that the terms in the exponent can be related to the identity for the difference of cubes:
\[ x^3 - y^3 = (x-y)(x^2 + xy + y^2) \]
Thus, we can rewrite the exponent as:
\[ x^3 - y^3 + y^3 - z^3 + z^3 - x^3 \]

### Step 6: Simplify the Exponent
When we add these terms together, they cancel out:
\[ (x^3 - y^3) + (y^3 - z^3) + (z^3 - x^3) = 0 \]

### Step 7: Final Result
Thus, we have:
\[ a^0 = 1 \]

### Conclusion
The expression simplifies to \(1\).

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