Question

( x^l / x^m )^l2–ml+m2 × ( x^m / x^n )^m2–mn+n2 × ( x^n / x^l )^n2–nl+l2



( don't give unnecessary answers ) ( give correct answer only ) ​

Answer 1

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

We know,

$\green{\boxed{ \bf \: \: {a}^{x} \div {a}^{y} = {a}^{x - y} }}$

$\green{\boxed{ \bf \: \: {a}^{x} \times {a}^{y} = {a}^{x + y} }}$

$\green{\boxed{ \bf \: {x}^{0} = 1\: }}$

$\green{\boxed{ \bf \: {( {x}^{m} )}^{n} \: = \: {x}^{mn} }}$

and

$\green{\boxed{ \bf \: (x - y)( {x}^{2} + xy + {y}^{2}) = {x}^{3} - {y}^{3}\: }}$

Consider,

$\red{\bf :\longmapsto\: {\bigg(\dfrac{ {x}^{l} }{ {x}^{m} } \bigg) }^{ {l}^{2} - lm + {m}^{2} }}$

$\rm \: = \: \: {\bigg( {x}^{(l - m)} \bigg) }^{ {l}^{2} - lm + {m}^{2} }$

$\rm \: = \: \: {\bigg( {x}\bigg) }^{ (l - m)({l}^{2} - lm + {m}^{2} )}$

$\rm \: = \: \: {\bigg( {x}\bigg) }^{({l}^{3} -{m}^{3} )}$

$\red{\bf :\implies\: {\bigg(\dfrac{ {x}^{l} }{ {x}^{m} } \bigg) }^{ {l}^{2} - lm + {m}^{2} } = {x}^{( {l}^{3} - {m}^{3})} - - - - (1)}$

Consider,

$\green{\bf :\longmapsto\: {\bigg(\dfrac{ {x}^{m} }{ {x}^{n} } \bigg) }^{ {m}^{2} - mn + {n}^{2} }}$

$\rm \: = \: \: {\bigg( {x}^{(m - n)} \bigg) }^{ {m}^{2} - mn + {n}^{2} }$

$\rm \: = \: \: {\bigg( {x} \bigg) }^{(m - n)( {m}^{2} - mn + {n}^{2}) }$

$\rm \: = \: \: {\bigg( {x}\bigg) }^{({m}^{3} -{n}^{3} )}$

$\green{\bf :\implies\: {\bigg(\dfrac{ {x}^{m} }{ {x}^{n} } \bigg) }^{ {m}^{2} - mn + {n}^{2} } = {x}^{( {m}^{3} - {n}^{3})} - - - - (2)}$

Consider

$\pink{\bf :\longmapsto\: {\bigg(\dfrac{ {x}^{n} }{ {x}^{l} } \bigg) }^{ {n}^{2} - nl \: + {l}^{2} }}$

$\rm \: = \: \: {\bigg( {x}^{(n - l)} \bigg) }^{ {n}^{2} - ln + {l}^{2} }$

$\rm \: = \: \: {\bigg( {x} \bigg) }^{(n - l)( {n}^{2} - ln + {l}^{2})}$

$\rm \: = \: \: {\bigg( {x}\bigg) }^{({n}^{3} -{l}^{3} )}$

$\pink{\bf :\implies\: {\bigg(\dfrac{ {x}^{n} }{ {x}^{l} } \bigg) }^{ {n}^{2} - nl + {l}^{2} } = {x}^{( {n}^{3} - {l}^{3})} - - - - (3)}$

Now,

Consider,

$\blue{\bf :\longmapsto\:{\bigg(\dfrac{ {x}^{l} }{ {x}^{m} } \bigg) }^{ {l}^{2} - lm \: + {m}^{2} } \times {\bigg(\dfrac{ {x}^{m} }{ {x}^{n} } \bigg) }^{ {m}^{2} - nm \: + {n}^{2} } \times {\bigg(\dfrac{ {x}^{n} }{ {x}^{l} } \bigg) }^{ {n}^{2} - nl \: + {l}^{2} }}$

$\rm \: = \: \: {x}^{ {l}^{3} - {m}^{3} } \times {x}^{ {m}^{3} - {n}^{3} } \times {x}^{ {l}^{3} - {m}^{3} }$

$\rm \: = \: \: {x}^{ {l}^{3} - {m}^{3} + {m}^{3} - {n}^{3} + {n}^{3} - {l}^{3} }$

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

$\rm \: = \: \: 1$

Hence,

$\blue{\bf :\longmapsto\:{\bigg(\dfrac{ {x}^{l} }{ {x}^{m} } \bigg) }^{ {l}^{2} - lm \: + {m}^{2} } \times {\bigg(\dfrac{ {x}^{m} }{ {x}^{n} } \bigg) }^{ {m}^{2} - nm \: + {n}^{2} } \times {\bigg(\dfrac{ {x}^{n} }{ {x}^{l} } \bigg) }^{ {n}^{2} - nl \: + {l}^{2} } = 1}$

More Identities to know:

• (a + b)² = a² + 2ab + b²

• (a - b)² = a² - 2ab + b²

• a² - b² = (a + b)(a - b)

• (a + b)² = (a - b)² + 4ab

• (a - b)² = (a + b)² - 4ab

• (a + b)² + (a - b)² = 2(a² + b²)

• (a + b)³ = a³ + b³ + 3ab(a + b)

• (a - b)³ = a³ - b³ - 3ab(a - b)