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.
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.