Question

(PEC)Determine o valor da expressão abaixo:
3^x+2 + 3^x+1/3^x-1

a) 36
b) 48
c) 27
d) 81

Answer 1

Resposta:

O valor da expressão apresentada pela questão equivale a 36.

⠀

Vamos determinar o valor da seguinte expressão:

\begin{gathered}\quad\quad\quad\Large\begin{array}{l}\sf\dfrac{3^{\,x\,+\,2}+3^{\,x\,+\,1}}{3^{\,x\,-\,1}}=~?\end{array}\\\\\end{gathered}

3

x−1

3

x+2

+3

x+1

= ?

→ A primeira coisa que percebemos é que podemos colocar o fator comum em evidência no numerador. Partindo da propriedade de potências de mesma base na multiplicação

\begin{gathered}\quad\quad\quad\quad \ \large\boxed{\begin{array}{l}\sf a^{b\,+\,c}~\Leftrightarrow~a^b\cdot a^c\end{array}}\\\\\end{gathered}

a

b+c

⇔ a

b

⋅a

c

, observe que:

\begin{gathered}\\\large\begin{array}{l}\quad\quad\quad\ \ \sf\dfrac{3^{\,x\,+\,2}+3^{\,x\,+\,1}}{3^{\,x\,-\,1}}\\\\\sf\iff~~~\dfrac{3^{\,x\,+\,1\,+\,1}+3^{\,x\,+\,1}}{3^{\,x\,-\,1}}\\\\\sf\iff~~~\dfrac{3^{\,x\,+\,1}\cdot3^1+3^{\,x\,+\,1}}{3^{\,x\,-\,1}}\\\\\sf\iff~~~\dfrac{3^{\,x\,+\,1}\cdot(3+1)}{3^{\,x\,-\,1}}\\\\\sf\iff~~~\dfrac{3^{\,x\,+\,1}\cdot4}{3^{\,x\,-\,1}}\end{array}\\\\\end{gathered}

3

x−1

3

x+2

+3

x+1

⟺

3

x−1

3

x+1+1

+3

x+1

⟺

3

x−1

3

x+1

⋅3

1

+3

x+1

⟺

3

x−1

3

x+1

⋅(3+1)

⟺

3

x−1

3

x+1

⋅4

Dessa forma colocamos o fator 3ˣ⁺¹ em evidência! Agora pela propriedade de potências de mesma base na divisão

\begin{gathered}\quad\quad\quad\quad\quad\large\boxed{\begin{array}{l}\sf \dfrac{~a^b}{~a^c}~\Leftrightarrow~a^{b\,-\,c}\end{array}}\\\\\end{gathered}

a

c

a

b

⇔ a

b−c

, obtemos:

\begin{gathered}\\\large\begin{array}{l}\sf\iff~~~\dfrac{3^{\,x\,+\,1}\cdot4}{3^{\,x\,-\,1}}\\\\\sf\iff~~~3^{\,x\,+\,1\,-\,(x\,-\,1)}\cdot4\\\\\sf\iff~~~3^{\,x\,+\,1\,-\,x\,+\,1}\cdot4\\\\\sf\iff~~~3^{\,0\,+\,2}\cdot4\\\\\sf\iff~~~3^{\,2}\cdot4\\\\\sf\iff~~~9\cdot4\\\\\sf\quad\!\therefore\quad~~\boldsymbol{\boxed{\sf36}}\end{array}\\\\\end{gathered}

⟺

3

x−1

3

x+1

⋅4

⟺ 3

x+1−(x−1)

⋅4

⟺ 3

x+1−x+1

⋅4

⟺ 3

0+2

⋅4

⟺ 3

2

⋅4

⟺ 9⋅4

∴

36

R: assim o valor simplificado daquela expressão inicial é 36.

Explicação passo-a-passo:

tmj