#1**+10 **

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

CPhill
Dec 10, 2014

#1**+10 **

Best Answer

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

CPhill
Dec 10, 2014