Question Number 115018 by bemath last updated on 23/Sep/20

If 9^x +9^(−x)  = 3^(2+x) +3^(2−x)  −20, then   27^x +27^(−x)  =?

Answered by bobhans last updated on 23/Sep/20

⇒ (3^x )^2 +(3^(−x) )^2 =9.3^x +9.3^(−x) −20  ⇒ set 3^x =a ∧3^(−x) =b  ⇒(a+b)^2 −2=9(a+b)−20  ⇒(a+b)^2 −9(a+b)+18=0  ⇒ a+b = 3 ∨ a+b = 6  now 27^x +27^(−x) =a^3 +b^3 =(a+b)^3 −3ab(a+b)  where ab = 1  ⇔ a^3 +b^3 =(a+b){(a+b)^2 −3}  for a+b = 3 ⇒ a^3 +b^3  = 3×6=18                 for a+b = 6 ⇒a^3 +b^3  = 6×33=198

Answered by PRITHWISH SEN 2 last updated on 23/Sep/20

let 3^x =a  a^2 +(1/a^2 ) = 9(a+(1/a))−20  t^2 −9t+18=0  {let t=(a+(1/a))}  t=3,6  27^x +27^(−x) = (a^3 +(1/a^3 ))= 3^3 −3.3=18 when t=3                       = 6^3 −3.6=216−18=198