$\left(\frac{x^a}{x^b}\right)^{\frac{1}{ab}} \cdot \left(\frac{x^b}{x^c}\right)^{\frac{1}{bc}} \cdot \left(\frac{x^c}{x^a}\right)^{\frac{1}{ca}} =$ $x$

Concept & Formula This problem evaluates your ability to combine the laws of indices with algebraic fraction splitting. The primary index laws used here are: $$\frac{x^m}{x^n} = x^{m-n}$$ $$(x^m)^n = x^{mn}$$ The crucial algebraic concept is splitting a fraction with a common denominator into two separate terms: $$\frac{m - n}{mn} = \frac{m}{mn} - \frac{n}{mn} = \frac{1}{n} - \frac{1}{m}$$ Step-by-step Solution Let's break down the expression one term at a time. First, simplify the inner fractions using the division law: 1. First term: $\frac{x^a}{x^b} = x^{a-b}$ 2. Second term: $\frac{x^b}{x^c} = x^{b-c}$ 3. Third term: $\frac{x^c}{x^a} = x^{c-a}$ Next, apply the outer fractional exponent to each base: 1. First term: $(x^{a-b})^{\frac{1}{ab}} = x^{\frac{a-b}{ab}}$ 2. Second term: $(x^{b-c})^{\frac{1}{bc}} = x^{\frac{b-c}{bc}}$ 3. Third term: $(x^{c-a})^{\frac{1}{ca}} = x^{\frac{c-a}{ca}}$ Now, use the fraction splitting rule on the exponents: 1. First exponent: $\frac{a-b}{ab} = \frac{a}{ab} - \frac{b}{ab} = \frac{1}{b} - \frac{1}{a}$ 2. Second exponent: $\frac{b-c}{bc} = \frac{b}{bc} - \frac{c}{bc} = \frac{1}{c} - \frac{1}{b}$ 3. Third exponent: $\frac{c-a}{ca} = \frac{c}{ca} - \frac{a}{ca} = \frac{1}{a} - \frac{1}{c}$ Multiply the terms together by adding their simplified exponents: $$x^{(\frac{1}{b} - \frac{1}{a}) + (\frac{1}{c} - \frac{1}{b}) + (\frac{1}{a} - \frac{1}{c})}$$ All terms in the exponent cancel out: $\frac{1}{b} - \frac{1}{b} = 0$ $-\frac{1}{a} + \frac{1}{a} = 0$ $\frac{1}{c} - \frac{1}{c} = 0$ The sum of the exponents is $0$. $$x^0 = 1$$ Exam Strategy & Shortcut **Value Putting Technique:** Since this is a generalized cyclic identity, the result will hold true for any non-zero values of $a, b,$ and $c$. Let $a=1, b=1, c=1$. The expression simplifies to $(\frac{1}{1})^1 \cdot (\frac{1}{1})^1 \cdot (\frac{1}{1})^1 = 1$. This bypasses all the complex fraction arithmetic and secures the answer instantly. Common Pitfall Students often get intimidated by the fractional exponents and attempt to find a massive common denominator ($abc$) for the entire expression right at the beginning. This creates unnecessary complexity. Splitting the fractions first is much faster and less error-prone. Final Answer **Therefore, the correct answer is 1.**