Given:
[tex]\sf{\dots\implies{\dfrac{{9}^{n} \times {3}^{2} \times({{3}^{\frac{- n}{2} })}^{- 2} -27^2}{ {3}^{3m}\times{2}^{3}}}}[/tex]
[tex]\rule{80mm}{1pt}[/tex]
What are asked to do?
We need to simply [tex]\sf{\frac{{9}^{n} \times {3}^{2} \times({{3}^{\frac{- n}{2} })}^{- 2} -27^2}{ {3}^{3m}\times{2}^{3}}}[/tex].
[tex]\rule{80mm}{1pt}[/tex]
Solution:
[tex]\sf{\dots\implies{\dfrac{{9}^{n} \times {3}^{2} \times({{3}^{\frac{- n}{2} })}^{- 2} -(27)^2}{ {3}^{3m}\times{2}^{3}}}}[/tex]
[tex]\sf{\dots\implies{\dfrac{{3}^{2n} \times {3}^{2} \times({{3}^{\frac{ \cancel{- }n}{ \cancel2} })}^{ \cancel{- 2}} -(3^{3} )^{2} }{ {3}^{3m}\times{2}^{3}}}}[/tex]
[tex]\sf{\dots\implies{\dfrac{{3}^{2n} \times {3}^{2} \times({{3)}^{n}} -(3 )^{6} }{ {3}^{3m}\times{2}^{3}}}}[/tex]
Since the base (3) is same so just add the exponents of multiple one.
[tex]\sf{\dots\implies{\dfrac{{3}^{(2n + 2 + n)}-(3 )^{6} }{ {3}^{3m}\times{2}^{3}}}}[/tex]
[tex]\sf{\dots\implies{\dfrac{{3}^{(3n + 2)}-(3 )^{6} }{ {3}^{3m}\times{2}^{3}}}}[/tex]
[tex]\sf{\dots\implies{\dfrac{{3}^{(3n + 2)}-(27 )^{2} }{ {3}^{3m}\times{2}^{3}}}}[/tex]
[tex]\sf{\dots\implies{\dfrac{{3}^{(3n + 2)}-( {3}^{2} \times {9}^{2} )}{ {3}^{3m}\times{2}^{3}}}}[/tex]
Take 3² as common.
[tex]\sf{\dots\implies{\dfrac{ {3}^{2}(({3)}^{3n}-9^{2})}{ {3}^{3m} \times 8}}}[/tex]
Solve the powers.
[tex]\sf{\dots\implies{\dfrac{ 9({27}^{n}-81)}{ {27}^{m} \times 8}}}[/tex]
Again take 27 as common.
[tex]\sf{\dots\implies{\dfrac{ 9 \times 27({1}^{n}-3)}{ {27}^{m} \times 8}}}[/tex]
