Question

Anything except answer will be reported.

Answer 1

$\sf \rightarrow \quad \dfrac{ { \bigg( 1 + \dfrac{p}{q} \bigg)}^{ \frac{p}{p - q} } { \bigg(1 - \dfrac{q}{p} \bigg)}^{ \frac{q}{p - q} } }{{ \bigg( 1 + \dfrac{q}{p} \bigg)}^{ \frac{p}{p - q} } { \bigg( \dfrac{p}{q} - 1 \bigg)}^{ \frac{q}{p - q} }} \\ \\ \sf \rightarrow \quad \bigg(\dfrac{ \frac{p + q}{q} }{ \frac{p + q}{p} } \bigg)^{ \frac{p}{p - q} } \times \bigg(\frac{ \frac{p - q}{p} }{ \frac{p - q}{q} } \bigg)^{ \frac{q}{p - q} } \\ \\ \sf \rightarrow \quad \bigg(\frac{ \cancel{p + q}}{q} \times \frac{p}{ \cancel{p + q}} \bigg)^{ \frac{p}{p - q} } \times \bigg( \frac{ \cancel{p - q}}{p} \times \frac{q}{ \cancel{p - q}} \bigg)^{ \frac{q}{p - q} } \\ \\ \sf \rightarrow \quad \bigg( \frac{p}{q} \bigg)^{ \frac{p}{p - q} } \times \bigg(\frac{q}{p} \bigg)^{ \frac{q}{p - q} } \\ \\ \sf \rightarrow \quad \dfrac{1}{ \bigg(\dfrac{p}{q} \bigg)^{ - \frac{p}{p - q} } } \times \bigg(\frac{q}{p} \bigg)^{ \frac{q}{p - q} } \\ \\ \sf \rightarrow \quad \bigg( \frac{q}{p} \bigg)^{ - \frac{p}{p - q} } \times \bigg( \frac{q}{p} \bigg)^{ \frac{q}{p - q} } \\ \\ \sf \rightarrow \quad \bigg( \frac{q}{p} \bigg)^{ \frac{q}{p - q} - \frac{p}{p - q} } \\ \\ \sf \rightarrow \quad \bigg( \frac{q}{p} \bigg)^{ \frac{q - p}{p - q} } \\ \\ \sf \rightarrow \quad \bigg( \frac{q}{p} \bigg)^{ \frac{ - \cancel{(p - q)}}{ \cancel{p - q}} } \\ \\ \sf \rightarrow \quad \bigg(\frac{q}{p} \bigg)^{ - 1} \\ \\ \sf \rightarrow \quad \frac{p}{q}$

Formulas Used:

$\sf \quad a^{m} \times a^{n} = a^{m+n} \\ \\ \sf \quad a^{m} = \dfrac{1}{a^{-m}} \\ \\ \sf \quad \dfrac{a^{m}}{b^{m}} = \bigg( \dfrac{a}{b} \bigg)^{m}$

Hence, Proved.