Prove that:

$\Big(\dfrac{x^a}{x^b}\Big)^{a + b - c} \times \Big(\dfrac{x^b}{x^c}\Big)^{b + c - a} \times \Big(\dfrac{x^c}{x^a}\Big)^{c + a - b} = 1$

Given,

$\Big(\dfrac{x^a}{x^b}\Big)^{a + b - c} \times \Big(\dfrac{x^b}{x^c}\Big)^{b + c - a} \times \Big(\dfrac{x^c}{x^a}\Big)^{c + a - b} = 1$

Solving L.H.S :

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

Hence proved, $\Big(\dfrac{x^a}{x^b}\Big)^{a + b - c} \times \Big(\dfrac{x^b}{x^c}\Big)^{b + c - a} \times \Big(\dfrac{x^c}{x^a}\Big)^{c + a - b} = 1$.