If a, b and c are any positive real number then the value of `sqrt(a^(-1)b).sqrt(b^(-1)c).sqrt(c^(-1)a)` is

Options

A. `1//2`
B. `0`
C. `1`
D. `-1`

Answer

To solve the problem, we need to find the value of the expression:

\[ \sqrt{a^{-1}b} \cdot \sqrt{b^{-1}c} \cdot \sqrt{c^{-1}a} \]

### Step-by-step Solution:

1. **Rewrite the Expression:**
We can rewrite the expression using the property of square roots:
\[ \sqrt{a^{-1}b} = (a^{-1}b)^{1/2} \]
Therefore, the entire expression becomes:
\[ (a^{-1}b)^{1/2} \cdot (b^{-1}c)^{1/2} \cdot (c^{-1}a)^{1/2} \]

2. **Combine the Square Roots:**
We can combine the square roots:
\[ = \sqrt{(a^{-1}b)(b^{-1}c)(c^{-1}a)} \]

3. **Simplify the Inside of the Square Root:**
Now, we simplify the product inside the square root:
\[ (a^{-1}b)(b^{-1}c)(c^{-1}a) = a^{-1}b \cdot b^{-1}c \cdot c^{-1}a \]
When we multiply these, we can rearrange the terms:
\[ = \frac{b}{a} \cdot \frac{c}{b} \cdot \frac{a}{c} \]

4. **Cancel Out Terms:**
Notice that in the multiplication:
\[ = \frac{b}{a} \cdot \frac{c}{b} \cdot \frac{a}{c} = \frac{b \cdot c \cdot a}{a \cdot b \cdot c} = 1 \]

5. **Final Calculation:**
Therefore, we have:
\[ \sqrt{(a^{-1}b)(b^{-1}c)(c^{-1}a)} = \sqrt{1} = 1 \]

### Conclusion:
Thus, the value of the expression \(\sqrt{a^{-1}b} \cdot \sqrt{b^{-1}c} \cdot \sqrt{c^{-1}a}\) is:

\[ \boxed{1} \]