To solve this equation, we can first rewrite the expression in terms of positive exponents by using the property of negative exponents:

x^(-4/3) = 1/x^(4/3)

x^(-2/3) = 1/x^(2/3)

Now the equation becomes:

1/x^(4/3) - 1/x^(2/3) - 8 = 0

Now we can multiply every term by x^(4/3) in order to clear the fractions:

x - x^2 - 8x^(4/3) = 0

Rearranging the terms, we get:

x^2 + 8x^(4/3) - x = 0

Now let's introduce a substitution to simplify further. Let u = x^(1/3). Now our equation becomes:

u^6 + 8u^4 - u^3 = 0

This is a polynomial equation that can be factored as:

u^3(u^3 - 1) + 8u^4 = 0

u^3(u - 1)(u^2 + u + 1) + 8u^4 = 0

Now we substitute back u = x^(1/3):

x^(1/3)(x^(1/3) - 1)(x^(2/3) + x^(1/3) + 1) + 8x^(4/3) = 0

However, this equation cannot be further simplified in terms of elementary functions and requires numerical methods to find the solutions.