To find an equivalent form of the rational expression \( 27h^8 - 18h^5 + 12h^3 \), we can factor out the greatest common factor (GCF) from the terms.
- The coefficients are 27, -18, and 12. The GCF of these coefficients is 9.
- The terms all contain varying powers of \( h \). The lowest power of \( h \) in the expression is \( h^3 \).
Thus, we can factor out \( 9h^3 \):
\[ 27h^8 - 18h^5 + 12h^3 = 9h^3(3h^5 - 2h^2 + 1) \]
Now we can rewrite it in different forms. However, if we simplify further by factoring or combining terms while maintaining equivalent expressions, we need to verify the provided options:
- Option A: \( 9h^7 - 6h^4 \) does not represent the original expression.
- Option B: This says the rational expression has no equivalent forms, which is incorrect because we just factored it.
- Option C: \( 9h^8 - 6h^5 + 4h \) does not match the expression.
- Option D: \( 9h^7 - 6h^4 + 4 \) does not represent the original expression.
None of the provided options are correct. The simplest equivalent form based on our factoring could be stated as:
\[ 9h^3(3h^5 - 2h^2 + 1) \]
If you need to select one despite them being incorrect, you should note that no accurate equivalent representation was provided in your choices.