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Solve for n
$\frac{ {9}^{n} \times {3}^{2} \times {3}^{n} - (27 {)}^{n} }{( {3}^{ {3})^{5} } \times {2}^{3} }$

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Answer 1

n = 4

Step by step explanation

$\\\sf{~~~~\implies~~~~\dfrac{ {9}^{n} \times {3}^{2} \times {3}^{n} - (27 {)}^{n} }{( {3}^{ {3})^{5} } \times {2}^{3} } = \dfrac{1}{27} }$

$\sf{~~~~\implies~~~~\dfrac{ {3}^{2n} \times {3}^{2} \times {3}^{n} - (3 {)}^{3n} }{ {3}^{15} \times {2}^{3} } = \dfrac{1}{ {3}^{3} } }$

$\sf{~~~~\implies~~~~\dfrac{ {3}^{2n + 2 + n} - (3 {)}^{3n} }{ {3}^{15} \times {2}^{3} } = \dfrac{1}{ {3}^{3} } }$

$\sf{~~~~\implies~~~~\dfrac{ {3}^{3n + 2} - (3 {)}^{3n} }{ {3}^{15} \times {2}^{3} } = \dfrac{1}{ {3}^{3} } }$

$\sf{~~~~\implies~~~~\dfrac{ {3}^{3n }( {3}^{2} - 1) }{ {3}^{15} \times 8 } = \dfrac{1}{ 27} }$

$\sf{~~~~\implies~~~~\dfrac{ {3}^{3n }( 9 - 1) }{ {3}^{15} \times 8 } = \dfrac{1}{ 27} }$

$\sf{~~~~\implies~~~~\dfrac{ {3}^{3n } \times 8 }{ {3}^{15} \times 8 } = \dfrac{1}{ 27} }$

$\sf{~~~~\implies~~~~\dfrac{ {3}^{3n } }{ {3}^{15} } = \dfrac{1}{ 27} }$

$\sf{~~~~\implies~~~~{ {3}^{3n } } = \dfrac{ {3}^{15} }{ {3}^{3} } }$

$\sf{~~~~\implies~~~~{ {3}^{3n } } = {3}^{12} }$

$\sf{~~~~\implies~~~~{ 3n } = 12 }$

$\sf{~~~~\implies~~~~{ n } = 4 }$

$\huge{\frak{~~~~~~~~n = 4 }}$

• $\tiny{\begin{gathered}\begin{gathered}\\\\\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \bigstar \: \underline{\bf{}}\\ {\boxed{\begin{array}{c | c} \frac{ \: ~~~~~~~~~~\: \: \: \: \:\sf Laws \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }{ } &\frac{ \: ~~~~~~~~~~ \: \: \: \: \:\sf Example \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }{ }\\ \sf \bigstar{a}^{m} \times {a}^{n} = {a}^{m + n} & \sf {a}^{2} \times {a}^{3} = {a}^{2 + 3} = {a}^{6} \\ \\ \sf \bigstar{a}^{m} \div {a}^{n} = {a}^{m - n}& \sf {a}^{3} \div {a}^{2} = {a}^{3 - 2} = {a}^{1} \\ \\ \sf{\bigstar \: \: \: \: \: \: ( {a}^{m} ) ^{n} = {a}^{mn} } & \sf( {a}^{2} ) ^{3} = {a}^{2 \times 3} = {a}^{6} \\ \\ {\bigstar\sf a {}^{m} \times {n}^{m} = (ab) ^{m} } &\sf a {}^{2} \times {b}^{2} = (ab) ^{2}\\ \\ \sf\bigstar \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: {a}^{0} = 1& \sf {2}^{0} = 1 \: \: \: \: \\ \\ \sf \bigstar \: \: \: \: {\dfrac{ {a}^{m} }{ {b}^{m} }= \left( \dfrac{a}{b} \right) ^{m} }& \sf{\dfrac{ {a}^{2} }{ {b}^{2} }= \left( \dfrac{a}{b} \right) ^{2} }\\\\\bigstar~~~~~~~ \sf x^{\frac{m}{n} }=\sqrt[n]{x^m}\sf = (\sqrt[n]{x})^m & \sf x^{\frac{2}{3} }=\sqrt[3]{x^2} = (\sqrt[n]{x})^m\\ \\\\ \end{array}}}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}}$

Answer 2

$\huge \tt \: n = 4$