`((x^a)/(x^b))^(1//ab)xx((x^b)/(x^c))^(1//bc)xx((x^c)/(x^a))^(1//ca)=?`

Options

A. 1
B. `x^(1//abc)`
C. `x^(1//(ab+bc+ca))`
D. None of these

Answer

To solve the expression

\[ \left(\frac{x^a}{x^b}\right)^{\frac{1}{ab}} \cdot \left(\frac{x^b}{x^c}\right)^{\frac{1}{bc}} \cdot \left(\frac{x^c}{x^a}\right)^{\frac{1}{ca}} \]

we will simplify each term step by step.

### Step 1: Simplify Each Fraction
We can simplify each fraction inside the parentheses:

1. \(\frac{x^a}{x^b} = x^{a-b}\)
2. \(\frac{x^b}{x^c} = x^{b-c}\)
3. \(\frac{x^c}{x^a} = x^{c-a}\)

### Step 2: Apply the Exponents
Now we apply the exponents to each simplified fraction:

1. \(\left(\frac{x^a}{x^b}\right)^{\frac{1}{ab}} = (x^{a-b})^{\frac{1}{ab}} = x^{\frac{a-b}{ab}}\)
2. \(\left(\frac{x^b}{x^c}\right)^{\frac{1}{bc}} = (x^{b-c})^{\frac{1}{bc}} = x^{\frac{b-c}{bc}}\)
3. \(\left(\frac{x^c}{x^a}\right)^{\frac{1}{ca}} = (x^{c-a})^{\frac{1}{ca}} = x^{\frac{c-a}{ca}}\)

### Step 3: Combine the Terms
Now we can combine all the terms together:

\[ x^{\frac{a-b}{ab}} \cdot x^{\frac{b-c}{bc}} \cdot x^{\frac{c-a}{ca}} = x^{\left(\frac{a-b}{ab} + \frac{b-c}{bc} + \frac{c-a}{ca}\right)} \]

### Step 4: Simplify the Exponent
Now we need to simplify the exponent:

\[ \frac{a-b}{ab} + \frac{b-c}{bc} + \frac{c-a}{ca} \]

To combine these fractions, we need a common denominator, which is \(abc\):

1. \(\frac{(a-b)c}{abc}\)
2. \(\frac{(b-c)a}{abc}\)
3. \(\frac{(c-a)b}{abc}\)

Combining these gives:

\[ \frac{(a-b)c + (b-c)a + (c-a)b}{abc} \]

### Step 5: Expand and Simplify
Now we expand the numerator:

\[ (a-b)c + (b-c)a + (c-a)b = ac - bc + ab - ac + bc - ab = 0 \]

Thus, the entire expression simplifies to:

\[ x^{\frac{0}{abc}} = x^0 = 1 \]

### Final Answer
Therefore, the final answer is:

\[ \boxed{1} \]