x^(-4/3)-x^(-2/3)-8=0 let u = x^(-2/3) then we have
u^2 - u - 8 = 0 add 8 to both sides
u^2 - u = 8 complete the square on both sides
u^2 - u + 1/4 = 8.25 factor
(u - 1/2)^2 = 8.25 take both roots
u - 1/2 = ±√8.25 add 1/2 to both sides
u = ±√8.25 + 1/2
So u = √8.25 + 1/2 or u = -√8.25 + 1/2
Then, back-substituting, we have
x^(-2/3) = √8.25 + 1/2 raise both sides to the (-3) power
x*2 = (√8.25 + 1/2)^(-3)
Take both roots
x = ±√((√8.25 + 1/2)^(-3)) = ±(√8.25 + 1/2)^(-3/2) = about ± .161
The other solution, u = -√8.25 + 1/2 leads to a non-real answer
Here's a graph.........https://www.desmos.com/calculator/kkomxbtlbj
x^(-4/3)-x^(-2/3)-8=0 let u = x^(-2/3) then we have
u^2 - u - 8 = 0 add 8 to both sides
u^2 - u = 8 complete the square on both sides
u^2 - u + 1/4 = 8.25 factor
(u - 1/2)^2 = 8.25 take both roots
u - 1/2 = ±√8.25 add 1/2 to both sides
u = ±√8.25 + 1/2
So u = √8.25 + 1/2 or u = -√8.25 + 1/2
Then, back-substituting, we have
x^(-2/3) = √8.25 + 1/2 raise both sides to the (-3) power
x*2 = (√8.25 + 1/2)^(-3)
Take both roots
x = ±√((√8.25 + 1/2)^(-3)) = ±(√8.25 + 1/2)^(-3/2) = about ± .161
The other solution, u = -√8.25 + 1/2 leads to a non-real answer
Here's a graph.........https://www.desmos.com/calculator/kkomxbtlbj