Question

please send solution ​

Answer 1

$\displaystyle \sf{ \bold { Solution \: - }} \\ \\ \sf{ \bold { \star To \: Show \: - }} \\ \\ \sf{ \implies { \bold { \dfrac{ x^{p(q-r)} }{ x^{q(p-r)}} \div \dfrac{ x^{qr} }{ x^{pr} } = 1 }}} \\ \\ \sf{ \implies { \bold { \dfrac{ x^{(pq-pr)} }{ x^{(pq-qr)} } \div \dfrac{ x^qr }{ x^pr } }}} \\ \\ \sf{ \implies { \bold { x^{(pq - pr - pq + qr )} \times x^{(pq - pr )} }}} \\ \\ \sf{ \implies { \bold { x^{( qr - pr + pr - qr )} }}} \\ \\ \sf{ \bold { \implies { 1 }}} \\ \\ \sf{ Hence \: Shown }$

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Additional Information :

( x^ a ) × ( x^ b ) = x^( a + b)

( x^ a )/ ( x^ b ) = x^( a - b )

a^0 = 1

( x^ a ) ^ b = x^( ab )

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Answer 2

$\large\underline\blue{\bold{Given \:  Question :-  }}$

$\sf \:  Show \:that \: \dfrac{ {x}^{p(q - r)} }{ {x}^{q(p - r)} } \div \ {(\dfrac{ {x}^{q} }{ {x}^{p} }) }^{r} = 1$

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$\huge{AηsωeR} ✍$

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$\large\underline\blue{\bold{Given :-  }}$

$\sf \:  \dfrac{ {x}^{p(q - r)} }{ {x}^{q(p - r)} } \div \ {(\dfrac{ {x}^{q} }{ {x}^{p} }) }^{r}$

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$\large\underline\blue{\bold{To \: Show :-  }}$

• Value is 1

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$\large\underline\blue{\bold{ Identity\: Used :-  }}$

$\begin{gathered}(1)\:{\underline{\boxed{\bf{\blue{a^m\times{a^n}\:=\:a^{m\:+\:n}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(2)\:{\underline{\boxed{\bf{\purple{\dfrac{a^m}{a^n}\:=\:a^{m\:-\:n}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(3)\:{\underline{\boxed{\bf{\orange{\dfrac{1}{x^n}\:=\:x^{-n}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(4)\:{\underline{\boxed{\bf{\color{peru}{(a^m)^n\:=\:a^{m\times{n}}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(5)\:{\underline{\boxed{\bf{\red{ {x}^{0} = 1}}}}} \\ \end{gathered}$

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$\large\underline\purple{\bold{Solution :- }}$

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$\sf \:  \dfrac{ {x}^{p(q - r)} }{ {x}^{q(p - r)} } \div \ {(\dfrac{ {x}^{q} }{ {x}^{p} }) }^{r}$

$\sf \:  ⟼\dfrac{ {x}^{(pq - pr)} }{ {x}^{(qp - qr)} } \div \ { {x}^{(q - p)r} }$

$\sf \:  ⟼ {x}^{(pq - pr - pq + qr)} \div {x}^{qr - pr}$

$\sf \:  ⟼ {x}^{( \cancel{pq} - pr - \cancel{pq} + qr)} \div {x}^{qr - pr}$

$\sf \:  ⟼ {x}^{( - pr + qr)} \div {x}^{qr - pr}$

$\sf \:  ⟼ {x}^{( - pr + qr - (qr - pr))}$

$\sf \:  ⟼ {x}^{( - pr + qr - qr + pr)}$

$\sf \:  ⟼ {x}^{( \cancel{ - pr} + \cancel{qr} - \cancel{qr} + \cancel{pr})}$

$\sf \:  ⟼ {x}^{0} = 1$

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$\large{\boxed{\boxed{\bf{Hence, Proved}}}}$