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Question

Find \(š‘„\), if \(9 Ɨ 3^š‘„ = (27) ^{2š‘„āˆ’3}\)

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Solution

Finding the value of \(š‘„\)

Given,

\(9 Ɨ 3^š‘„ = (27)^{ 2š‘„ā€“3}\)

\(ā‡’ 3^2 Ɨ 3^š‘„ = (3 Ɨ 3 Ɨ 3)^{ 2š‘„ā€“3}\)

\(ā‡’ 3^{2+š‘„} = (3^{3})^{2š‘„āˆ’3}~~~ [āˆµ š‘Ž^{š‘š}Ɨ š‘Ž^{š‘›} = š‘Ž^{š‘š+š‘›}]\)

\(ā‡’ 3^{2+š‘„} = 3^{3(2š‘„āˆ’3)}\)

Since the bases are same, compare the powers on both sides,

\(āˆ“ 2 + š‘„ = 3(2š‘„ āˆ’ 3)\)

\(ā‡’ 2 + š‘„ = 6š‘„ āˆ’ 9\)

\(ā‡’ 6š‘„ āˆ’ š‘„ = 9 + 2\)

\(ā‡’ 5š‘„ = 11\)

\(ā‡’ š‘„ = \dfrac{11}{5}= 2\dfrac{1}{5}\)

Hence, the value of \(š‘„\) is \(\dfrac{11}{ 5}\) or \(2\dfrac{ 1}{5}\)

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