Variable binomials with negative fractional exponents

[Asci]

I have to solve the following equation: ((x+3)^(-2/3))-((x+3)^(-1/3))=6

How would I go about solving this? I graphed the equation on my calculator and calculated the intersection with y=6 and got -3.125, which checks out if you plug it in, but I don't know how to do the algebra to arrive at this answer.

 

[skeeter]

let u = (x + 3)^-1/3

u² - u = 6

u² - u - 6 = 0

(u - 3)(u + 2) = 0

two solutions, u = 3 and u = -2

working with both solutions one at a time ...

u = 3 ...
(x + 3)^-1/3 = 3
cube both sides ...
(x + 3)^-1 = 27
1/(x + 3) = 27
x + 3 = 1/27
x = 1/27 - 3 = -80/27

u = -2 ...
(x + 3)^-1/3 = -2
cube both sides ...
(x + 3)^-1 = -8
1/(x + 3) = -8
x + 3 = -1/8
x = -1/8 - 3 = -25/8

ah! the power of substitution!

 

[Asci]

I checked both of your answers, and they work out! Thanks a lot, I really appreciate it!