Solve the followings :
`((x^a)/(x^b))^(a + b) xx ((x^b)/(x^c))^(b+c) xx ((x^c)/(x^a))^(c+a)` is equal to :

Options

A. 1
B. 0
C. 2
D. None of these

Answer

To solve the expression

\[ \left(\frac{x^a}{x^b}\right)^{a+b} \cdot \left(\frac{x^b}{x^c}\right)^{b+c} \cdot \left(\frac{x^c}{x^a}\right)^{c+a} \]

we will simplify it step by step.

### Step 1: Simplifying Each Term
We start by simplifying each term in the expression.

1. **First Term**:
\[ \left(\frac{x^a}{x^b}\right)^{a+b} = \left(x^{a-b}\right)^{a+b} = x^{(a-b)(a+b)} \]

2. **Second Term**:
\[ \left(\frac{x^b}{x^c}\right)^{b+c} = \left(x^{b-c}\right)^{b+c} = x^{(b-c)(b+c)} \]

3. **Third Term**:
\[ \left(\frac{x^c}{x^a}\right)^{c+a} = \left(x^{c-a}\right)^{c+a} = x^{(c-a)(c+a)} \]

### Step 2: Combine the Terms
Now we combine all the simplified terms:

\[ x^{(a-b)(a+b)} \cdot x^{(b-c)(b+c)} \cdot x^{(c-a)(c+a)} \]

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

\[ x^{(a-b)(a+b) + (b-c)(b+c) + (c-a)(c+a)} \]

### Step 3: Expand the Exponents
Now, we will expand each term in the exponent:

1. **First Term**:
\[ (a-b)(a+b) = a^2 - b^2 \]

2. **Second Term**:
\[ (b-c)(b+c) = b^2 - c^2 \]

3. **Third Term**:
\[ (c-a)(c+a) = c^2 - a^2 \]

### Step 4: Combine the Expanded Terms
Now we combine these results:

\[ a^2 - b^2 + b^2 - c^2 + c^2 - a^2 \]

Notice that \( a^2 \) and \( -a^2 \) cancel out, \( b^2 \) and \( -b^2 \) cancel out, and \( c^2 \) and \( -c^2 \) cancel out. Therefore, we are left with:

\[ 0 \]

### Final Result
Thus, the entire expression simplifies to:

\[ x^0 = 1 \]

### Conclusion
The value of the given expression is:

\[ \boxed{1} \]

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