Finding the value of \(š„\)
Given,
\(9 Ć 3^š„ = (27)^{ 2š„ā3}\)
\(ā 3^2 Ć 3^š„ = (3 Ć 3 Ć 3)^{ 2š„ā3}\)
\(ā 3^{2+š„} = (3^{3})^{2š„ā3}~~~ [āµ š^{š}Ć š^{š} = š^{š+š}]\)
\(ā 3^{2+š„} = 3^{3(2š„ā3)}\)
Since the bases are same, compare the powers on both sides,
\(ā“ 2 + š„ = 3(2š„ ā 3)\)
\(ā 2 + š„ = 6š„ ā 9\)
\(ā 6š„ ā š„ = 9 + 2\)
\(ā 5š„ = 11\)
\(ā š„ = \dfrac{11}{5}= 2\dfrac{1}{5}\)
Hence, the value of \(š„\) is \(\dfrac{11}{ 5}\) or \(2\dfrac{ 1}{5}\)