Evaluate : ` sqrt(x^(a-b)) xx sqrt( x^(b-c)) xx sqrt( x^(c-a))`

Answer

To evaluate the expression \( \sqrt{x^{a-b}} \times \sqrt{x^{b-c}} \times \sqrt{x^{c-a}} \), we can follow these steps:

### Step 1: Rewrite the square roots as exponents
We know that \( \sqrt{x^m} = x^{m/2} \). Therefore, we can rewrite the expression:
\[ \sqrt{x^{a-b}} = x^{(a-b)/2}, \quad \sqrt{x^{b-c}} = x^{(b-c)/2}, \quad \sqrt{x^{c-a}} = x^{(c-a)/2} \]
So, the expression becomes:
\[ x^{(a-b)/2} \times x^{(b-c)/2} \times x^{(c-a)/2} \]

### Step 2: Combine the exponents
Using the property of exponents that states \( x^m \times x^n = x^{m+n} \), we can combine the exponents:
\[ x^{(a-b)/2 + (b-c)/2 + (c-a)/2} \]

### Step 3: Simplify the exponent
Now, we simplify the exponent:
\[ \frac{(a-b) + (b-c) + (c-a)}{2} \]
When we expand the numerator:
\[ (a - b + b - c + c - a) = 0 \]
Thus, we have:
\[ \frac{0}{2} = 0 \]

### Step 4: Substitute back into the expression
Now substituting back, we get:
\[ x^0 \]
According to the properties of exponents, any non-zero number raised to the power of 0 is equal to 1:
\[ x^0 = 1 \]

### Final Answer:
Thus, the value of the expression \( \sqrt{x^{a-b}} \times \sqrt{x^{b-c}} \times \sqrt{x^{c-a}} \) is:
\[ \boxed{1} \]