$\left( \frac{x^a}{x^b} ight)^{\frac{1}{ab}} imes \left( \frac{x^b}{x^c} ight)^{\frac{1}{bc}} imes \left( \frac{x^c}{x^a} ight)^{\frac{1}{ca}}$ =

Question

$\left( \frac{x^a}{x^b} ight)^{\frac{1}{ab}} 	imes \left( \frac{x^b}{x^c} ight)^{\frac{1}{bc}} 	imes \left( \frac{x^c}{x^a} ight)^{\frac{1}{ca}}$ =

Accepted Answer

To solve the given expression $\left( \frac{x^a}{x^b} \right)^{\frac{1}{ab}} \times \left( \frac{x^b}{x^c} \right)^{\frac{1}{bc}} \times \left( \frac{x^c}{x^a} \right)^{\frac{1}{ca}}$, we need to simplify each part step by step.

1. **Simplifying Each Term:**

- The first term is $\left( \frac{x^a}{x^b} \right)^{\frac{1}{ab}}$. This simplifies to:
     $$
     \left( x^{a-b} \right)^{\frac{1}{ab}} = x^{\frac{a-b}{ab}}
     $$

- The second term is $\left( \frac{x^b}{x^c} \right)^{\frac{1}{bc}}$. This simplifies to:
     $$
     \left( x^{b-c} \right)^{\frac{1}{bc}} = x^{\frac{b-c}{bc}}
     $$

- The third term is $\left( \frac{x^c}{x^a} \right)^{\frac{1}{ca}}$. This simplifies to:
     $$
     \left( x^{c-a} \right)^{\frac{1}{ca}} = x^{\frac{c-a}{ca}}
     $$

2. **Combining the Terms:**

Now, we multiply these simplified terms together:
   $$
   x^{\frac{a-b}{ab}} \times x^{\frac{b-c}{bc}} \times x^{\frac{c-a}{ca}}
   $$

Using the property of exponents that states $x^m \times x^n = x^{m+n}$, we combine the exponents:
   $$
   x^{\frac{a-b}{ab} + \frac{b-c}{bc} + \frac{c-a}{ca}}
   $$

3. **Simplifying the Exponent:**

To simplify the exponent, we find a common denominator, which is $abc$:
   $$
   \frac{a-b}{ab} = \frac{c(a-b)}{abc}, \quad \frac{b-c}{bc} = \frac{a(b-c)}{abc}, \quad \frac{c-a}{ca} = \frac{b(c-a)}{abc}
   $$

Adding these fractions:
   $$
   \frac{c(a-b) + a(b-c) + b(c-a)}{abc}
   $$

Simplifying the numerator:
   $$
   ca - cb + ab - ac + bc - ba = 0
   $$

Therefore, the exponent becomes:
   $$
   \frac{0}{abc} = 0
   $$

4. **Final Result:**

Since the exponent is 0, the entire expression simplifies to:
   $$
   x^0 = 1
   $$

Thus, the correct answer is Option 1: 1.

**Explanation for Incorrect Options:**

- **Option 2 (0):** The expression simplifies to $x^0$, which is 1, not 0.
- **Option 3 ($x^{abc}$):** This would imply the exponent is $abc$, but we showed the exponent simplifies to 0.
- **Option 4 (abc):** This is not a valid result for the expression, as it simplifies to a power of $x$, specifically $x^0$.

Therefore, the correct answer is indeed Option 1: 1.