Evaluate :

$\Big(\dfrac{x^q}{x^r}\Big)^{\dfrac{1}{qr}} \times \Big(\dfrac{x^r}{x^p}\Big)^{\dfrac{1}{rp}} \times \Big(\dfrac{x^p}{x^q}\Big)^{\dfrac{1}{pq}}$

Simplifying the expression :

$\Rightarrow \Big(\dfrac{x^q}{x^r}\Big)^{\dfrac{1}{qr}} \times \Big(\dfrac{x^r}{x^p}\Big)^{\dfrac{1}{rp}} \times \Big(\dfrac{x^p}{x^q}\Big)^{\dfrac{1}{pq}} \\[1em] = (x^{q - r})^{\dfrac{1}{qr}} \times (x^{r - p})^{\dfrac{1}{rp}} \times (x^{p - q})^{\dfrac{1}{pq}} \\[1em] = (x)^{\dfrac{q - r}{qr}} \times (x)^{\dfrac{r - p}{rp}} \times (x)^{\dfrac{p - q}{pq}} \\[1em] = x^{\Big(\dfrac{q - r}{qr} + \dfrac{r - p}{rp} + \dfrac{p - q}{pq}\Big)} \\[1em] = x^{\Big(\dfrac{p(q - r) + q(r - p) + r(p - q)}{pqr}\Big)} \\[1em] = x^{\Big(\dfrac{pq - pr + qr - qp + rp - rq}{pqr}\Big)} \\[1em] = x^{\Big(\dfrac{0}{pqr}\Big)} \\[1em] = x^0 \\[1em] = 1.$

Hence, $\Big(\dfrac{x^q}{x^r}\Big)^{\dfrac{1}{qr}} \times \Big(\dfrac{x^r}{x^p}\Big)^{\dfrac{1}{rp}} \times \Big(\dfrac{x^p}{x^q}\Big)^{\dfrac{1}{pq}} = 1$.