Question

please prove. .................

Answer 1

Here the given question is
${\frac{ {x}^{a} }{ {x}^{b} }}^{ \frac{1}{ab} } \times {\frac{ {x}^{b} }{ {x}^{c} }}^{ \frac{1}{bc} } \times {\frac{ {x}^{c} }{ {x}^{a} }}^{ \frac{1}{ca} }$
${ {x}^{a - b}}^{ \frac{1}{ab} } \times { {x}^{b - c}}^{ \frac{1}{bc} } \times { {x}^{c - a}}^{ \frac{1}{ca} }$
${x}^{ \frac{a - b}{ab} } \times {x}^{ \frac{b - c}{bc} } \times {x}^{ \frac{c - a}{ca} }$
${x}^{ \frac{a - b}{ab} +\frac{b - c}{bc} + \frac{c - a}{ca} }$
${x}^{ \frac{1}{a} - \frac{1}{b} +\frac{1}{b} - \frac{1}{c} + \frac{1}{c} - \frac{1}{a}}$

${x}^{0} = 1$
Hence proved.

Hope it helps.