Question

simplify the given indices​

Answer 1

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

The given expression is

$\rm \: {\bigg( {\bigg(a\bigg) }^{\dfrac{1}{x - y} } \bigg) }^{\dfrac{1}{x - z} }{\bigg( {\bigg(a \bigg) }^{\dfrac{1}{y - z} } \bigg) }^{\dfrac{1}{y - x} }{\bigg( {\bigg(a\bigg) }^{\dfrac{1}{z - x} } \bigg) }^{\dfrac{1}{z - y} }$

We know that,

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

So, using this identity, we get

$\rm \: {{\bigg(a\bigg) }^{\dfrac{1}{(x - y)(x - z)} }}{{\bigg(a \bigg) }^{\dfrac{1}{(y - z)(y - x)} }}{{\bigg(a\bigg) }^{\dfrac{1}{(z - x)(z - y)} }}$

We know,

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

So, using this property, we get

$\rm \: = \: {{\bigg(a\bigg) }^{\dfrac{1}{(x - y)(x - z)} + \dfrac{1}{(y - z)(y - x)} + \dfrac{1}{(z - x)(z - y)}}}$

$\rm \: = \: {{\bigg(a\bigg) }^{ - \dfrac{1}{(x - y)(z - x)} - \dfrac{1}{(y - z)(x - y)} - \dfrac{1}{(z - x)(y - z)}}}$

$\rm \: = \: {{\bigg(a\bigg) }^{ - \bigg( \dfrac{1}{(x - y)(z - x)} + \dfrac{1}{(y - z)(x - y)} + \dfrac{1}{(z - x)(y - z)} \bigg)}}$

$\rm \: = \: {{\bigg(a\bigg) }^{ - \: \dfrac{y - z + z - x + x - y}{(x - y)(y - z)(z - x)} }}$

$\rm \: = \: {{\bigg(a\bigg) }^{ - \: \dfrac{0}{(x - y)(y - z)(z - x)} }}$

$\rm \: = \: {a}^{0}$

$\rm \: = \: 1$

Additional Information :-

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

$\underbrace{\boxed{ \tt{ {x}^{0} \: = \: 1}}}$

$\underbrace{\boxed{ \tt{ {x}^{ - y} \: = \: \frac{1}{ {x}^{y} } }}}$