a)
\(\left(\frac{1}{3}\right)^n\cdot27^n=3^n\)
\(\Rightarrow\left(\frac{1}{3}\cdot27\right)^n=3^n\)
\(\Rightarrow9^n=3^n\)
\(\Rightarrow\left(3^2\right)^n=3^n\)
\(\Rightarrow3^{2n}=3^n\)
\(\Rightarrow2n=n\)
\(\Leftrightarrow n=0\)
Vậy \(n=0\)
d) Ta có:
\(6^{3-n}=216\)
\(\Rightarrow6^{3-n}=6^3\)
\(\Rightarrow3-n=3\)
\(\Rightarrow n=3-3\)
\(\Rightarrow n=0\)
Vậy \(n=0\)\(\text{ }\)
e) Ta có:
\(5^n+5^{n+2}=650\)
\(\Rightarrow5^n+5^n\cdot5^2=650\)
\(\Rightarrow5^n\cdot\left(1+5^2\right)=650\)
\(\Rightarrow5^n\cdot26=25\cdot26\)
\(\Rightarrow5^n=25\)
\(\Rightarrow5^n=5^2\)
\(\Rightarrow n=2\)
Vậy \(n=2\)
b) Ta có:
\(\frac{1}{2}\cdot2^n+4\cdot2^n=3^2\cdot2^5\)
\(\Rightarrow2^n\cdot\left(4+\frac{1}{2}\right)=9\cdot2^5\)
\(\Rightarrow2^n\cdot\frac{9}{2}=9\cdot2^5\)
\(\Rightarrow2^n=2^5\cdot9:\frac{9}{2}\)
\(\Rightarrow2^n=2^5\cdot2\)
\(\Rightarrow2^n=2^6\)
\(\Rightarrow n=6\)
Vậy \(n=6\)
c) Ta có:
\(\frac{64}{\left(-2\right)^n}=-32\)
\(\Rightarrow\frac{2^6}{\left(-2\right)^n}=\left(-2\right)^5\)
\(\Rightarrow2^n=2^6:\left(-2\right)^5\)
\(\Rightarrow2^n=2^6:2^5:\left(-1\right)\)
\(\Rightarrow2^n=2:\left(-1\right)\)
\(\Rightarrow2^n=-2\) (vô lí)
=> Không có giá trị n thỏa mãn.
Vậy không có giá trị n thỏa mãn.
\(\)
f) Ta có:
\(3^{n-1}+5\cdot3^{n-1}=162\)
\(\Rightarrow3^{n-1}\cdot\left(1+5\right)=162\)
\(\Rightarrow3^{n-1}\cdot6=27\cdot6\)
\(\Rightarrow3^{n-1}=27\)
\(\Rightarrow3^{n-1}=3^3\)
\(\Rightarrow n-1=3\)
\(\Rightarrow n=4\)
Vậy \(n=4\)
