a) 27^n : 3^n = 9
(27 : 3)^n = 9
9^n = 9
=> n = 1
b) 25/5^n = 5
5^n = 25 : 5
5^n = 5
=> n = 1
c) 81/(-3)^n = -243
(-3)^n = -243 : 81
(-3)^n = -3
=> n = 1
d) 1/2 . 2^n + 4 . 2^n = 9 . 2^5
2^n . (1/2 + 4) = 9 . 32
2^n . 9/2 = 288
2^n = 288 : 9/2
2^n = 64
2^n = 2^6
=> n = 6
thank you for doing your homework well !
\(a)27^n\div3^n=9\)
\(\Rightarrow\left(27\div3\right)^n=9\)
\(\Rightarrow9^n=9\Rightarrow n=1\)
\(b)\frac{25}{5^n}=5\Rightarrow25=5^{n+1}\Rightarrow5^2=5^{n+1}\Rightarrow2=n+1\Rightarrow n=1\)
\(c)\frac{81}{\left(-3\right)^n}=-243\Rightarrow\frac{81}{\left(-3\right)^n}=\left(-3\right)^5\)
\(\Rightarrow81=\left(-3\right)^{n+5}\Rightarrow\left(-3\right)^4=\left(-3\right)^{n+5}\)
\(\Rightarrow4=n+5\Rightarrow n=-1\)
\(d)\frac{1}{2}.2^n+4.2^n=9.2^5\Rightarrow2^n\left(\frac{1}{2}+4\right)=9.2^5\)
\(\Rightarrow2^n.\frac{9}{2}=9.2^5\Rightarrow2^n=9.2^5.\frac{2}{9}\Rightarrow2^n=\frac{9.2^5.2}{9}=2^6\)
\(\Rightarrow n=6\)
