If $a$, $b$, $c$ are real numbers, then the value of $$ \sqrt{a^{-1}b} \cdot \sqrt{b^{-1}c} \cdot \sqrt{c^{-1}a} $$ is

### Concept & Formula This problem requires the application of negative exponent rules and radical properties. The rule for negative exponents is $x^{-n} = \frac{1}{x^n}$. The multiplication rule for radicals with the same index is $\sqrt{x} \cdot \sqrt{y} = \sqrt{xy}$. ### Step-by-Step Solution * **Step 1:** Rewrite the terms with negative exponents as positive denominators. $\sqrt{a^{-1}b} = \sqrt{\frac{b}{a}}$ $\sqrt{b^{-1}c} = \sqrt{\frac{c}{b}}$ $\sqrt{c^{-1}a} = \sqrt{\frac{a}{c}}$ * **Step 2:** Combine the individual square roots into a single large square root by multiplying the terms inside. $$ \sqrt{\frac{b}{a}} \cdot \sqrt{\frac{c}{b}} \cdot \sqrt{\frac{a}{c}} = \sqrt{\frac{b}{a} \cdot \frac{c}{b} \cdot \frac{a}{c}} $$ * **Step 3:** Cancel out the common terms in the numerator and denominator. The $a$ in the numerator cancels the $a$ in the denominator. The $b$ in the numerator cancels the $b$ in the denominator. The $c$ in the numerator cancels the $c$ in the denominator. This leaves: $$ \sqrt{1} $$ * **Step 4:** Evaluate the final square root. $\sqrt{1} = 1$ ### Exam Strategy & Shortcut **Visual Cancellation:** Once you recognize the structure $a^{-1}b$, you should instantly visualize it as $b/a$. Looking at the chain $b/a \cdot c/b \cdot a/c$, the cyclic pattern guarantees that every variable appears exactly once in the numerator and once in the denominator. They all cancel to $1$, making the square root of $1$ just $1$. This can be solved completely in your head in 5 seconds. ### Common Pitfall A trivial mistake is forgetting that $\sqrt{1}$ equals $1$ and instead trying to match the cyclic nature of the variables to option (a) $abc$, assuming the variables must be preserved in the answer. Always follow the arithmetic to its conclusion. ### Final Answer Therefore, the correct answer is **1**.