The value of `((x^(a))/(x^(b)))^((1)/(ab)) xx ((x^(b))/(x^(c)))^((1)/(bc)) xx ((x^(c))/(x^(a)))^((1)/(ca)) ` is equal to

Options

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

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: Simplifying Each Fraction
The first term can be simplified as follows:

\[ \frac{x^a}{x^b} = x^{a-b}. \]

Thus,

\[ \left(\frac{x^a}{x^b}\right)^{\frac{1}{ab}} = \left(x^{a-b}\right)^{\frac{1}{ab}} = x^{\frac{a-b}{ab}}. \]

### Step 2: Simplifying the Second Term
Now, for the second term:

\[ \frac{x^b}{x^c} = x^{b-c}. \]

So,

\[ \left(\frac{x^b}{x^c}\right)^{\frac{1}{bc}} = \left(x^{b-c}\right)^{\frac{1}{bc}} = x^{\frac{b-c}{bc}}. \]

### Step 3: Simplifying the Third Term
Next, for the third term:

\[ \frac{x^c}{x^a} = x^{c-a}. \]

Therefore,

\[ \left(\frac{x^c}{x^a}\right)^{\frac{1}{ca}} = \left(x^{c-a}\right)^{\frac{1}{ca}} = x^{\frac{c-a}{ca}}. \]

### Step 4: Combining All Terms
Now we can combine all the simplified terms:

\[ x^{\frac{a-b}{ab}} \cdot x^{\frac{b-c}{bc}} \cdot x^{\frac{c-a}{ca}}. \]

Using the property of exponents that states \(x^m \cdot x^n = x^{m+n}\), we can write:

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

### Step 5: Finding a Common Denominator
To combine the exponents, we need a common denominator. The common denominator for \(ab\), \(bc\), and \(ca\) is \(abc\).

Rewriting each term with the common denominator:

\[ \frac{a-b}{ab} = \frac{(a-b)c}{abc}, \quad \frac{b-c}{bc} = \frac{(b-c)a}{abc}, \quad \frac{c-a}{ca} = \frac{(c-a)b}{abc}. \]

### Step 6: Combining the Exponents
Now we can combine the fractions:

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

### Step 7: Simplifying the Numerator
Expanding the numerator:

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

### Step 8: Final Expression
Thus, we have:

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

### Final Answer
The value of the given expression is

\[ \boxed{1}. \]