To determine an equivalent form of the given rational expression \( 27h^8 - 18h^5 + 12h^3h \), we first simplify it.
The term \( 12h^3h \) can be simplified to \( 12h^4 \). So, we rewrite the expression:
\[ 27h^8 - 18h^5 + 12h^4 \]
Next, we look for a common factor in the expression \( 27h^8 - 18h^5 + 12h^4 \). The greatest common factor (GCF) of the coefficients \( 27, -18, \) and \( 12 \) is \( 9 \), and the common factor in terms of \( h \) is \( h^4 \).
Factoring \( 9h^4 \) out of each term gives:
\[ 9h^4(3h^4 - 2h + 1.33) \]
However, for convenience, let’s factor it in a simpler way:
- \( 27h^8 \) becomes \( 9h^4 \cdot 3h^4 \)
- \( -18h^5 \) becomes \( 9h^4 \cdot (-2h) \)
- \( 12h^4 \) becomes \( 9h^4 \cdot \frac{4}{3} \)
Our factored form is:
\[ 9h^4(3h^4 - 2h + \frac{4}{3}) \]
As we break it down, we can analyze the options given:
- \( 9h^7 - 6h^4 \)
- \( 9h^8 - 6h^5 + 4h \)
- \( 9h^7 - 6h^4 + 4 \)
None of these options match our final factored outcome directly.
To evaluate them further:
- \( 9h^7 - 6h^4 \) does not have a term to represent \( 27h^8 \) or \( -18h^5 + 12h^4 \).
- \( 9h^8 - 6h^5 + 4h \) includes a wrong coefficient for \( h^8 \) and an incorrect arrangement for \( h^5\) and \( h^4 \).
- \( 9h^7 - 6h^4 + 4 \) is also incorrect based on dimensionality and coefficient validations.
Therefore, the conclusion is:
This rational expression does not have any equivalent forms as per the options provided.