$\left(\frac{x^a}{x^b}\right)^{(a+b)} \cdot \left(\frac{x^b}{x^c}\right)^{(b+c)} \cdot \left(\frac{x^c}{x^a}\right)^{(c+a)} =$ $x$

Concept & Formula This problem relies on three fundamental algebraic rules working together: the division law of indices, the power law of indices, and the difference of squares identity. The core laws of indices are: $$\frac{x^m}{x^n} = x^{m-n}$$ $$(x^m)^n = x^{mn}$$ The algebraic identity is: $$(a - b)(a + b) = a^2 - b^2$$ Step-by-step Solution First, simplify the base of each fractional term using the division law of indices: 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 exponent to each simplified base using the power law of indices and the difference of squares identity: 1. First term: $(x^{a-b})^{(a+b)} = x^{(a-b)(a+b)} = x^{a^2 - b^2}$ 2. Second term: $(x^{b-c})^{(b+c)} = x^{(b-c)(b+c)} = x^{b^2 - c^2}$ 3. Third term: $(x^{c-a})^{(c+a)} = x^{(c-a)(c+a)} = x^{c^2 - a^2}$ Now, multiply the three resulting terms together by adding their exponents: $$x^{(a^2 - b^2) + (b^2 - c^2) + (c^2 - a^2)}$$ Notice that every term in the exponent cancels out perfectly: $a^2 - a^2 = 0$ $-b^2 + b^2 = 0$ $-c^2 + c^2 = 0$ The entire exponent simplifies to $0$, leaving us with: $$x^0 = 1$$ Exam Strategy & Shortcut **Cyclic Symmetry Method:** When an algebraic expression perfectly cycles through variables (e.g., $a \rightarrow b \rightarrow c \rightarrow a$), all terms will naturally balance and cancel each other out. You can instantly deduce that the exponent sum will be $0$. Alternatively, substitute $a=1, b=1, c=1$. The entire expression becomes $1^2 \cdot 1^2 \cdot 1^2 = 1$. This takes 2 seconds and guarantees accuracy. Common Pitfall A frequent error is trying to expand $(\frac{x^a}{x^b})^{(a+b)}$ as $\frac{(x^a)^{(a+b)}}{(x^b)^{(a+b)}}$. While mathematically valid, it creates a massive fraction with 6 terms that is highly prone to calculation errors. Always simplify the inner parenthesis first! Final Answer **Therefore, the correct answer is 1.**