Solve `(8^x+27^x)/(12^x+18^x)=7/6`

Answer

`x = -1 or 1`

Explanation

`(8^(x) + 27^(x))/(12^(x) + 18^(x) )=(7)/(6)`
or `(8)^(x)/(12)^(x) ((1 +(27//8)^x))/((1 +(18//12)^x))= (7)/(6)`
or `((2)/(3))^(x)((1 +(3//2)^(3x))/(1 +(3//2)^x))= (7)/(6)`
Let `((3)/(2))^(x) = t` . Then
`(1 +t^(3))/(t(1 + t)) = (7)/(6)`
or `((1 +t)(t^(2) + 1 - t))/(t(1 + t)) = (7)/(6)`
or `(t^(2)+ 1 - t)/(t) = (7)/(6) (because t + 1 ne` 0)
or `6t^(2) - 13t + 6 = 0`
or `(2t - 3)(3t-2) = 0`
`rArr t = (2)/(3) or (3)/(2)`
`rArr x = - 1 or 1` .