smaplify the following \( \frac{27 x^{3}-8}{27 x^{3}+18 x+12} \)

To simplify \( \frac{27 x^{3}-8}{27 x^{3}+18 x+12} \), we can factor the numerator and the denominator. 1. The numerator \( 27x^3 - 8 \) can be recognized as a difference of cubes, which factors as: \[ (3x - 2)(9x^2 + 6x + 4) \] 2. The denominator \( 27x^3 + 18x + 12 \) can be factored by looking for common terms or using polynomial long division, though it doesn't factor nicely like the numerator. However, evaluating for simplifications, since there are no common factors between the numerator and the denominator, the expression remains: \[ \frac{27x^3-8}{27x^3+18x+12} \quad \text{which simplifies no further.} \] So, the simplified form is \( \frac{(3x - 2)(9x^2 + 6x + 4)}{27x^3 + 18x + 12} \).