If a = xy^{p - 1}, b = xy^{q - 1} and c = xy^{r - 1}, prove that

a^{q - r}.b^{r - p}.c^{p - q} = 1.

Given,

a = xy^{p - 1}, b = xy^{q - 1} and c = xy^{r - 1}.

Substituting value of a, b and c in L.H.S. of a^{q - r}.b^{r - p}.c^{p - q} = 1 we get,

⇒ (xy^{p - 1})^{q - r}.(xy^{q - 1})^{r - p}.(xy^{r - 1})^{p - q}

⇒ (xy)^{(p - 1)(q - r)}.(xy)^{(q - 1)(r - p)}.(xy)^{(r - 1)(p - q)}

⇒ (xy)^{(p - 1)(q - r) + (q - 1)(r - p) + (r - 1)(p - q)}

⇒ (xy)^{pq - pr - q + r + qr - qp - r + p + rp - rq - p + q}

⇒ (xy)^{p - p - q + q + r - r + pq - qp - pr + rp + qr - rq}

⇒ (xy)⁰

⇒ 1.

Hence,prove that a^{q - r}.b^{r - p}.c^{p - q} = 1.