Question

pls solve it.. ...........​

Answer 1

GIVEN :–

$\\ \implies \sf { \left( \dfrac{ {x}^{b} }{ {x}^{c} } \right)}^{ \frac{1}{bc} } \times { \left( \dfrac{ {x}^{c} }{ {x}^{a} } \right)}^{ \frac{1}{ca} } \times { \left( \dfrac{ {x}^{a} }{ {x}^{b} } \right)}^{ \frac{1}{ab} }$

TO FIND :–

• Value of given expression = ?

SOLUTION :–

$\\ \: \: = \: \: \sf { \left( \dfrac{ {x}^{b} }{ {x}^{c} } \right)}^{ \frac{1}{bc} } \times { \left( \dfrac{ {x}^{c} }{ {x}^{a} } \right)}^{ \frac{1}{ca} } \times { \left( \dfrac{ {x}^{a} }{ {x}^{b} } \right)}^{ \frac{1}{ab} }$

• Using identity –

$\\ \longrightarrow \: \: \sf \dfrac{ {a}^{b} }{ {a}^{c} } = {a}^{b - c}$

$\\ \: \: = \: \: \sf { ( {x}^{b - c})}^{ \frac{1}{bc} } \times {({x}^{c - a} )}^{ \frac{1}{ca} } \times {( {x}^{a - b} )}^{ \frac{1}{ab} }$

• Using identity –

$\\ \longrightarrow \: \: \sf ({a}^{b})^{c} = {a}^{bc}$

• So that –

$\\ \: \: = \: \: \sf \left({ {x}^{( \frac{b - c}{bc})}} \right) \left({ {x}^{ ( \frac{c - a}{ca})}} \right) \left({ {x}^{( \frac{a - b }{ab})}} \right)$

$\\ \: \: = \: \: \sf \left({ {x}^{ \frac{1}{c} - \frac{1}{b} }} \right) \left( { {x}^{ \frac{1}{a} - \frac{1}{c} }} \right) \left( { {x}^{ \frac{1}{b} - \frac{1}{a} }} \right)$

• Using identity –

$\\ \longrightarrow \: \: \sf { {a}^{b} }.{ {a}^{c} } = {a}^{b + c}$

• So that –

$\\ \: \: = \: \: \sf \left( { {x}^{ \frac{1}{c} - \frac{1}{b} + \frac{1}{a} - \frac{1}{c} + \frac{1}{b} - \frac{1}{a} }} \right)$

$\\ \: \: = \: \: \sf {x}^{0}$

$\\ \: \: = \: \: \sf1$

• Hence , Option (D) is correct .