Question

(ITA) - Sendo $2^{x-3} +2^{x-4} = 2^{x-2}-2^{x-1}+14$, então x é:
a) Um número par
b) Um quadrado perfeito
c) Um número primo
d) Um múltiplo de 3
e) Zero

Eu sei o valor de x, quero saber como se resolve.

Answer 1

Olá Equisbacon.


Organizando e desenvolvendo a equação.


$\mathsf{2^{x-3}+2^{x-4}=2^{x-2}-2^{x-1}+14}\\\\\mathsf{2^x\cdot2^{-3}+2^x\cdot2^{-4}=2^x\cdot2^{-2}-2^x\cdot2^{-1}+14}\\\\\mathsf{2^x\cdot\dfrac{1}{2^3}+2^x\cdot\dfrac{1}{2^4}=2^x\cdot\dfrac{1}{2^2}-2^x\cdot\dfrac{1}{2}+14}\\\underline{\qquad\qquad\qquad\qquad}\\\\\mathsf{2^x=y}\\\\\underline{\qquad\qquad\qquad\qquad}\\\\\mathsf{\dfrac{y}{2^3}+\dfrac{y}{2^4}=\dfrac{y}{2^2}-\dfrac{y}{2}+14~\cdot(2^4)}\\\\\mathsf{2y+y=4y-8y+14\cdot2^4}\\\\\mathsf{3y+4y=14\cdot2^4}\\\\\mathsf{7y=14\cdot2^4}$

$\mathsf{y=\dfrac{\diagup\!\!\!\!14\cdot2^4}{\diagup\!\!\!\!7}}\\\\\mathsf{y=2\cdot2^4}\\\\\mathsf{y=2^5}\\\\\\\mathsf{2^x=y}\\\\\mathsf{2^x=2^5}\\\\\boxed{\mathsf{x=5}}$

Resposta (c)

Dúvidas? comente.