Question

Tìm n \(\in\) Z

a) \(\dfrac{1}{9}.27^n=3^n\)

b) \(3^{-2}.3^4.3^n=3^7\)

c) \(2^{-1}.2^n+4.2^n=9.2^5\)

d) \(32^{-n}.16^{-n}=2048\)

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Answer 1

a) \(\dfrac{1}{9}.27^n=3^n\)

\(\dfrac{1}{3^2}.3^{3n}=3^n\\ \Rightarrow3^{3n-2}=3^n\\ \Rightarrow3n-2=n\\ \Rightarrow n=1\)

b) \(3^{-2}.3^4.3^n=3^7\)

\(\dfrac{1}{3^2}.3^4.3^n=3^7\\ \Rightarrow3^{n+2}=3^7\Rightarrow n+2=7\\ \Rightarrow n=5\)

c) \(2^{-1}.2^n+4.2^n=9.2^5\)

\(\dfrac{1}{2}.2^n+4.2^n=9.2^5\\ \Rightarrow2^n\left(\dfrac{1}{2}+4\right)=9.2^5\\ \Rightarrow2^{n-1}.9=9.2^5\\ \Rightarrow n-1=5\\ \Rightarrow n=6\)

d) \(32^{-n}.16^{-n}=2048\)

\(\dfrac{1}{2^n.16^n}.16^n=2^{11}=\dfrac{1}{2^n}=2^{11}\\ \Rightarrow2^n.2^{11}=1\\ \Rightarrow2^{n+11}=2^0\\ \Rightarrow n+11=0\\ \Rightarrow n=-11\)

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