Simplify :

$\Big(\dfrac{x^a}{x^b}\Big)^{a^2 + ab + b^2} \times \Big(\dfrac{x^b}{x^c}\Big)^{b^2 + bc + c^2} \times \Big(\dfrac{x^c}{x^a}\Big)^{c^2 + ca + a^2}$

Simplifying the expression :

$\Rightarrow \Big(\dfrac{x^a}{x^b}\Big)^{a^2 + ab + b^2} \times \Big(\dfrac{x^b}{x^c}\Big)^{b^2 + bc + c^2} \times \Big(\dfrac{x^c}{x^a}\Big)^{c^2 + ca + a^2} \\[1em] = (x^{a - b})^{a^2 + ab + b^2} \times (x^{b - c})^{b^2 + bc + c^2} \times (x^{c - a})^{c^2 + ca + a^2} \\[1em] = x^{(a - b)(a^2 + ab + b^2)} \times x^{(b - c)(b^2 + bc + c^2)} \times x^{(c - a)(c^2 + ca + a^2)} \\[1em] = x^{a^3 - b^3} \times x^{b^3 - c^3} \times x^{c^3 - a^3} \\[1em] = x^{a^3 - b^3 + b^3 - c^3 + c^3 - a^3} \\[1em] = x^0 \\[1em] = 1.$

Hence, $\Big(\dfrac{x^a}{x^b}\Big)^{a^2 + ab + b^2} \times \Big(\dfrac{x^b}{x^c}\Big)^{b^2 + bc + c^2} \times \Big(\dfrac{x^c}{x^a}\Big)^{c^2 + ca + a^2}$ = 1.