If 9n.32.3n−(27)n(3m.2)3=3−3\dfrac{9^n.3^2.3^n - (27)^n}{(3^m.2)^3} = 3^{-3}(3m.2)39n.32.3n−(27)n=3−3.
Show that : m - n = 1.
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Given,
⇒9n.32.3n−(27)n(3m.2)3=3−3⇒(32)n.32.3n−(33)n(3m)3.(2)3=3−3⇒32n.32.3n−33n=3−3.(3m)3.(2)3⇒9.32n+n−33n=3−3.33m.8⇒9.33n−33n=8.33m−3⇒33n(9−1)=8.33m−3⇒8.33n=8.33(m−1)⇒33n=33(m−1)⇒3n=3(m−1)⇒n=m−1⇒m−n=1.\Rightarrow \dfrac{9^n.3^2.3^n - (27)^n}{(3^m.2)^3} = 3^{-3} \\[1em] \Rightarrow \dfrac{(3^2)^n.3^2.3^n - (3^3)^n}{(3^m)^3.(2)^3} = 3^{-3} \\[1em] \Rightarrow 3^{2n}.3^2.3^n - 3^{3n} = 3^{-3}.(3^m)^3.(2)^3 \\[1em] \Rightarrow 9.3^{2n + n} - 3^{3n} = 3^{-3}.3^{3m}.8 \\[1em] \Rightarrow 9.3^{3n} - 3^{3n} = 8.3^{3m - 3} \\[1em] \Rightarrow 3^{3n}(9 - 1) = 8.3^{3m - 3} \\[1em] \Rightarrow 8.3^{3n} = 8.3^{3(m - 1)} \\[1em] \Rightarrow 3^{3n} = 3^{3(m - 1)} \\[1em] \Rightarrow 3n = 3(m - 1) \\[1em] \Rightarrow n = m - 1 \\[1em] \Rightarrow m - n = 1.⇒(3m.2)39n.32.3n−(27)n=3−3⇒(3m)3.(2)3(32)n.32.3n−(33)n=3−3⇒32n.32.3n−33n=3−3.(3m)3.(2)3⇒9.32n+n−33n=3−3.33m.8⇒9.33n−33n=8.33m−3⇒33n(9−1)=8.33m−3⇒8.33n=8.33(m−1)⇒33n=33(m−1)⇒3n=3(m−1)⇒n=m−1⇒m−n=1.
Hence, proved that m - n = 1.
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