Answer :

DᴀʀᴋPᴀʀᴀᴅᴏx

[tex]{ \qquad\qquad\huge\underline{{\sf Answer}}} [/tex]

Here we go ~

[tex]\qquad \sf  \dashrightarrow \: \dfrac{((243 {)}^{2n/5} ) \sdot( {3}^{2n + 1}) }{ ({9}^{n + 1}) \sdot( {3}^{2(n - 2)} )}[/tex]

[tex]\qquad \sf  \dashrightarrow \: \dfrac{(3{)}^{5(2n/5)} ) \sdot( {3}^{2n + 1}) }{ ({3 }^{2(n + 1)}) \sdot( {3}^{2(n - 2)} )}[/tex]

[tex]\qquad \sf  \dashrightarrow \: \dfrac{(3{)}^{2n} \sdot( {3}^{2n + 1}) }{ ({3 }^{(2n + 2)}) \sdot( {3}^{(2n - 4)} )}[/tex]

let's break it up :

  • [tex] \sf  3 {}^{2n} = (3 {}^{2n -4} ) \sdot(3 {}^{4} )[/tex]
  • [tex] \sf  3 {}^{(2n + 1)} = (3 {}^{2n -4} ) \sdot(3 {}^{5} )[/tex]
  • [tex] \sf  3 {}^{(2n + 2)} = (3 {}^{2n -4} ) \sdot(3 {}^{6} )[/tex]

now let's take [tex]{ \sf {3}^{(2n-4)}} [/tex] common here ~

[tex]\qquad \sf  \dashrightarrow \: \dfrac{(3{)}^{(2n - 4)} (3 {}^{4} \sdot{3}^{5}) }{ {(3 )}^{(2n - 4)}(3 {}^{6} \sdot1)}[/tex]

[tex]\qquad \sf  \dashrightarrow \: \dfrac{{} (3 {}^{4} \sdot{3}^{5}) }{ (3 {}^{6} \sdot1)}[/tex]

[tex]\qquad \sf  \dashrightarrow \: \dfrac{{} 3 {}^{9} }{ 3 {}^{6} }[/tex]

[tex]\qquad \sf  \dashrightarrow \: 3 {}^{3} [/tex]

[tex]\qquad \sf  \dashrightarrow \: 27[/tex]

Hope008

Answer:

  • 27

Step-by-step explanation:

[tex]\sf \cfrac{243^{\frac{2n}{5}}\cdot \:3^{2n+1}}{9^{n+1}\cdot \:3^{2\left(n-2\right)}}[/tex]

[tex]\sf 9^{n+1}\cdot \:3^{2\left(n-2\right)}[/tex]

Simplify:-

  • [tex]\sf 3^{4n-2}[/tex]

[tex]\sf \cfrac{3^{2n}\cdot \:3^{2n+1}}{\boxed{\bf 3^{4n-2}}}[/tex]

Now, Factor:-

  • [tex]\sf 243^{\frac{2n}{5}}[/tex]
  • [tex]= \boxed{\bf 3^{2n}}[/tex]

[tex]\sf \cfrac{3^{2n}\cdot \:3^{2n+1}}{3^{4n-2}}[/tex]

Now, let's simplify:-

  • [tex]\sf \cfrac{3^{2n}}{3^{4n-2}}[/tex]
  • [tex]=\boxed{\bf 3^{-2n+2}}[/tex]
  • [tex]\sf 3^{-2n+2}\cdot \:3^{2n+1}[/tex]

Simplify:-

  • [tex]\sf 3^{-2n+2}\cdot \:3^{2n+1}[/tex]

Apply the exponent rule:-

  • [tex]\sf 3^{-2n+2+2n+1}[/tex]
  • [tex]\sf 3^3[/tex]
  • [tex]\sf 27[/tex]

Therefore, your answer is 27.

______________________

Hope this helps!

Have a great day!

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