[tex]\displaystyle\bf\\a)\\\left(\frac{2}{3}\right)^n=\frac{16}{81}=\frac{2^4}{3^4}=\left(\frac{2}{3}\right)^4~\implies~n=4\\\\\\b)\\\left(\frac{1}{7}\right)^n=\frac{1}{343}=\frac{1}{7^3}=\left(\frac{1}{7}\right)^3~\implies~n=3\\\\\\c)\\\left(1\frac{1}{2}\right)^n=3\frac{375}{1000}\\\\\\\left(\frac{3}{2}\right)^n=\frac{3\times1000+375}{1000}\\\\\\\left(\frac{3}{2}\right)^n=\left(\frac{3375}{1000}\right)^{(125}=\left(\frac{27}{8}\right)=\left(\frac{3}{2}\right)^3\implies n=3[/tex]
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[tex]\displaystyle\bf\\d)\\\left(2\frac{1}{2}\right)^n=\frac{125}{8}\\\\\left(\frac{2\times2+1}{2}\right)^n=\frac{5^3}{2^3}\\\\\left(\frac{5}{2}\right)^n=\left(\frac{5}{2}\right)^3\\\\\implies~n=3\\\\e)\\\left(\frac{1}{3^2}\right)^n=\left(\frac{1}{729}\right)\\\\\left(\frac{1}{3}\right)^{2n}=\left(\frac{1}{3^6}\right)\\\\\left(\frac{1}{3}\right)^{2n}=\left(\frac{1}{3}\right)^6\\\\\implies~2n=6\implies n=\frac{6}{2} \implies n=3[/tex]
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[tex]\displaystyle\bf\\f)\\\left(\frac{1}{8}\right)^n=\left(\frac{1}{4^3}\right)\\\\\left(\frac{1}{2^3}\right)^n=\left(\frac{1}{\Big(2^2\Big)^3}\right)\\\\\\\left(\frac{1}{2}\right)^{3n}=\left(\frac{1}{2}\right)^6\implies3n=6\implies n=2[/tex]
