Question

Sabendo que 3^t (elevado a t) + 3^-t = 5
calcule 27^t + 27^-t

Answer 1

3ᵗ + 3⁻ᵗ = 5.

Elavando - se tudo ao cubo

(3ᵗ + 3⁻ᵗ)³ = 5³.

(3ᵗ)³+3.(3ᵗ)(3⁻ᵗ)(3ᵗ+3⁻ᵗ) + (3⁻ᵗ)³ = 5³

27ᵗ + 3.1.5 + 27⁻ᵗ = 5³

27ᵗ+27⁻ᵗ = 5³ - 15

27ᵗ+27⁻ᵗ = 125 - 15

= 110

Espero que te ajude!

Bons estudos!

Answer 2

Expressão algébrica :

$\mathsf{3^t+3^{-t}~=~5 }$

, $\mathsf{27^t+27^{-t}~=~? }$

Eleve ambos os membros ao quadrado , na Expressão dada :

$\mathsf{\Big(3^t+3^{-t} \Big)^2~=~5^2 }$

$\mathsf{\Big(3^t\Big)^2+2.\Big(3^t\Big).\Big(3^{-t}\Big)+\Big(3^{-t} \Big)^2~=~25 }$

$\mathsf{\Big(3^{2t}\Big)+2.1+\Big(3^{-2t}\Big)~=~25 }$

$\mathsf{3^{2t}+3^{-2t}~=~25-2 }$

$\mathsf{3^{2t}+3^{-2t}~=~23 }$

Agora vamos multiplicar ambos os membros por: $\mathsf{3^{t}+3^{-t} }$

$\mathsf{\Big(3^{2t}+3^{-2t}\Big)\Big(3^t+3^{-t} \Big)~=~23.\Big(3^t+3^{-t} \Big) }$

$\mathsf{3^{2t}.3^t+3^{2t}.3^{-t}+3^{-2t}.3^{t}+3^{-2t}.3^{-t}~=~23.5 }$

$\mathsf{3^{2t+t}+3^{2t-t}+3^{-2t+t}+3^{-2t-t}~=~115 }$

$\mathsf{3^{3t}+3^{t}+3^{-t}+3^{-3t}~=~115 }$

$\mathsf{\Big(3^3\Big)^t+\underbrace{3^t+3^{-t}}_{5}+\Big(3^3\Big)^{-t}~=~115 }$

$\mathsf{27^t+27^{-t}~=~115-5 }$

$\boxed{\mathsf{27^t+27^{-t}~=~110 }}}}$>>>>Resposta