Question

simplify the expression (x^a/x^-b)^a-b (x^b/x^-c)^b-c (x^c/x^-a)^c-a

please give the full explanation of the answer it's a humble request....​

Answer 1

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\: {\bigg(\dfrac{ {x}^{a} }{ {x}^{ - b} } \bigg) }^{a - b} {\bigg(\dfrac{ {x}^{b} }{ {x}^{ - c} } \bigg) }^{b - c} {\bigg(\dfrac{ {x}^{c} }{ {x}^{ - a} } \bigg) }^{c - a}$

Let first simply each term individually,

Consider,

$\rm :\longmapsto\: {\bigg(\dfrac{ {x}^{a} }{ {x}^{ - b} } \bigg) }^{a - b}$

We know that

$\underbrace{\boxed{ \tt{ \frac{ {x}^{m} }{ {x}^{n} } \: = \: {x}^{m \: - \: n} }}}$

Using this result, we get

$\rm \:  =  \:  \: {\bigg( {x}^{a - ( - b)} \bigg) }^{a - b}$

$\rm \:  =  \:  \: {\bigg( {x}^{a + b} \bigg) }^{a - b}$

We know,

$\underbrace{\boxed{ \tt{ {( {x}^{m} )}^{n} \: = \: {x}^{mn} }}}$

So, using this we get

$\rm \:  =  \:  \: {(x)}^{(a + b)(a - b)}$

$\rm \:  =  \:  \: {x}^{ {a}^{2} - {b}^{2} }$

Now, Consider,

$\rm :\longmapsto\: {\bigg(\dfrac{ {x}^{b} }{ {x}^{ - c} } \bigg) }^{b - c}$

We know that

$\underbrace{\boxed{ \tt{ \frac{ {x}^{m} }{ {x}^{n} } \: = \: {x}^{m \: - \: n} }}}$

So, using this identity we get

$\rm \:  =  \:  \: {\bigg( {x}^{b - ( - c)} \bigg) }^{b - c}$

$\rm \:  =  \:  \: {\bigg( {x}^{b + c} \bigg) }^{b - c}$

We know that

$\underbrace{\boxed{ \tt{ {( {x}^{m} )}^{n} \: = \: {x}^{mn} }}}$

So, using this, we get

$\rm \:  =  \:  \: {(x)}^{(b+c)(b - c)}$

$\rm \:  =  \:  \: {x}^{ {b}^{2} - {c}^{2} }$

Now, Consider

$\rm :\longmapsto\: {\bigg(\dfrac{ {x}^{c} }{ {x}^{ - a} } \bigg) }^{c - a}$

We know that,

$\underbrace{\boxed{ \tt{ \frac{ {x}^{m} }{ {x}^{n} } \: = \: {x}^{m \: - \: n} }}}$

So, using this result, we get

$\rm \:  =  \:  \: {\bigg( {x}^{c - ( - a)} \bigg) }^{c - a}$

$\rm \:  =  \:  \: {\bigg( {x}^{c + a} \bigg) }^{c - a}$

We know that,

$\underbrace{\boxed{ \tt{ {( {x}^{m} )}^{n} \: = \: {x}^{mn} }}}$

So, using this result, we get

$\rm \:  =  \:  \: {(x)}^{(c+a)(c - a)}$

$\rm \:  =  \:  \: {x}^{ {c}^{2} - {a}^{2} }$

Now, Consider the given expression

$\rm :\longmapsto\: {\bigg(\dfrac{ {x}^{a} }{ {x}^{ - b} } \bigg) }^{a - b} {\bigg(\dfrac{ {x}^{b} }{ {x}^{ - c} } \bigg) }^{b - c} {\bigg(\dfrac{ {x}^{c} }{ {x}^{ - a} } \bigg) }^{c - a}$

can be rewritten as using above,

$\rm \:  =  \:  \: {x}^{ {a}^{2} - {b}^{2} } \times {x}^{ {b}^{2} - {c}^{2} } \times {x}^{ {c}^{2} - {a}^{2} }$

We know that,

$\underbrace{\boxed{ \tt{ {x}^{m} \: \times \: {x}^{n} \: = \: {x}^{m \: + \: n} }}}$

So, using this result, we get

$\rm \:  =  \:  \: {x}^{ {a}^{2} - {b}^{2} + {b}^{2} - {c}^{2} + {c}^{2} - {a}^{2} }$

$\rm \:  =  \:  \: {x}^{0}$

$\rm \:  =  \:  \: 1$

Hence,

$\underbrace{\boxed{ \bf{ \:{\bigg(\dfrac{ {x}^{a} }{ {x}^{ - b} } \bigg) }^{a - b} \: {\bigg(\dfrac{ {x}^{b} }{ {x}^{ - c} } \bigg) }^{b - c} \: {\bigg(\dfrac{ {x}^{c} }{ {x}^{ - a} } \bigg) }^{c - a} \: = \: 1}}}$