Question

(U. Passo Fundo - RS) Simplificando-se a expressão
$\frac{ {3}^{x + 2} + {3}^{x + 1} + {3}^{x - 1} + {3}^{x - 2} }{ {3}^{x} + {3}^{x - 1} + {3}^{x - 2} }$
tem-se:
a)3-x
b)1/3
c)9
d)1/9
e)112/13

Answer 1

$\textbf{Algumas propriedades de potencia\c{c}\~ao:}\\\\ \mathrm{\Rightarrow a^m.a^n=a^{m+n}\ \ \bigg\|\ \ \Rightarrow \dfrac{a^m}{a^n}=a^{m-n}}\\\\ \textbf{Resolu\c{c}\~ao:}\\\\ \mathrm{\dfrac{3^{x+2}+3^{x+1}+3^{x-1}+3^{x-2}}{3^x+3^{x-1}+3^{x-2}}\ \to\ \dfrac{3^x.9+3^x.3+\dfrac{3^x}{3}+\dfrac{3^x}{9}}{3^x+\dfrac{3^x}{3}+\dfrac{3^x}{9}}=}\\\\ \mathrm{=\dfrac{12.3^x+\dfrac{3.3^x+3^x}{9}}{\dfrac{9.3^x+3.3^x+3^x}{9}}=\dfrac{\dfrac{108.3^x+4.3^x}{9}}{\dfrac{13.3^x}{9}}=\dfrac{112.3^x}{9}.\dfrac{9}{13.3^x}=\boxed{\mathbf{\dfrac{112}{13}}}}$