Question

sir Ch exponent
sir please find fast​

Answer 1

Given Question :-

Simplify :

$\rm :\longmapsto\:\dfrac{ {6}^{2n + 3} - {36}^{n + 2} }{ {\bigg[ {(216)}^{n + 1} \bigg]}^{\dfrac{2}{3} } }$

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

Given expression is

$\rm :\longmapsto\:\dfrac{ {6}^{n + 3} - {36}^{n + 2} }{ {\bigg[ {(216)}^{n + 1} \bigg]}^{\dfrac{2}{3} } }$

can be rewritten as

$\rm \:  =  \: \dfrac{ {6}^{2n + 3} - {(6 \times 6)}^{n + 2} }{ {\bigg[ {(6 \times 6 \times 6)}^{n + 1} \bigg]}^{\dfrac{2}{3} } }$

$\rm \:  =  \: \dfrac{ {6}^{2n + 3} - {( {6}^{2})}^{n + 2} }{ {\bigg[ {( {6}^{3} )}^{n + 1} \bigg]}^{\dfrac{2}{3} } }$

We know,

$\red{\boxed{ \tt{ \: {( {a}^{m} )}^{n} \: = \: {a}^{mn} \: }}}$

So, using this, we get

$\rm \:  =  \: \dfrac{ {6}^{2n + 3} - {6}^{2(n + 2)} }{ {\bigg[ {6}^{3(n + 1)} \bigg]}^{\dfrac{2}{3} } }$

$\rm \:  =  \: \dfrac{ {6}^{2n + 3} - {6}^{2(n + 2)} }{ {{6}^{2(n + 1)}} }$

$\rm \:  =  \: \dfrac{ {6}^{2n + 3} - {6}^{2n + 4} }{ {{6}^{2n +2}} }$

$\rm \:  =  \: \dfrac{ {6}^{2n + 3}}{ {6}^{2n + 2} } - \dfrac{ {6}^{2n + 4} }{ {6}^{2n + 2} }$

We know,

$\begin{gathered}\:{\underline{\boxed{\bf{\blue{a^m\div{a^n}\:=\:a^{m\: - \:n}\:}}}}} \\ \end{gathered}$

So, using this, we get

$\rm \:  =  \: {6}^{(2n + 3) - (2n + 2)} - {6}^{(2n + 4) - (2n + 2)}$

$\rm \:  =  \: {6}^{2n + 3 - 2n - 2} - {6}^{2n + 4 - 2n - 2}$

$\rm \:  =  \: 6 - {6}^{2}$

$\rm \:  =  \: 6 - 36$

$\rm \:  =  \: - 30$

Hence,

$\rm \implies\:\boxed{ \tt{ \: \dfrac{ {6}^{2n + 3} - {36}^{n + 2} }{ {\bigg[ {(216)}^{n + 1} \bigg]}^{\dfrac{2}{3} } } = - 30 \: }}$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

More to know :-

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

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

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

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

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

Answer 2

Question:-

$\sf \frac{6 {}^{2n + 3} - (36) {}^{n + 2} }{[(216) {}^{n + 1} ] {}^{ \frac{2}{3} } }$

• As we know that, 6³ = 216.

$\longmapsto \sf \frac{6 {}^{2n + 3} - (36) {}^{n + 2} }{[6 {}^{n + 1}] {}^{2} }$

$\longmapsto \sf \frac{6 {}^{2n + 3} }{6 {}^{2n + 2} } - \frac{6 {}^{2n + 4} }{6 {}^{2n + 2} }$

$\longmapsto \sf 6 - 36 = - 30$

Answer:-

$\sf \huge \mathfrak \green{ = - 30}$