Marissa was asked to create an equivalent form of the rational expression 27h8−18h5+12h3h . Which of the following is a correct equivalent form?(1 point)

Question

Responses

This rational expression does not have any equivalent forms.
This rational expression does not have any equivalent forms.

9h8−6h5+4h
9 h superscript 8 baseline minus 6 h superscript 5 baseline plus 4 h

9h7−6h4+4
9 h superscript 7 baseline minus 6 h superscript 4 baseline plus 4

9h7−6h4

Answer 1

To simplify the given expression $27h^8-18h^5+12h^{3}h$, we first need to rewrite the last term and then factor out the greatest common factor (GCF).

1. Rewrite the last term: $$12h^{3}h=12h^{3+1}=12h^4$$ Now, the expression is: $$27h^8-18h^5+12h^4$$

2. Identify the GCF: The GCF of the coefficients $27$, $18$, and $12$ is $9$. For the variable $h$, the smallest exponent in all terms is $h^4$.

3. Factor out the GCF: $$9h^4(3h^4-2h+\frac{4}{3})$$ We now re-arrange this to see if we can get to match any of the options given.

4. Alternatively, let's factor out $9h^4$ directly: $$27h^8=9h^4\cdot 3h^4$$ $$-18h^5=9h^4\cdot (-2h)$$ $$12h^4=9h^4\cdot \frac{4}{3}$$ Thus, combining gives: $$=9h^4(3h^4-2h+\frac{4}{3})$$

From this analysis, we see that there is a common factor of $9h^4$.

Now, let’s look at your options:

• Option 1: "This rational expression does not have any equivalent forms." - Not true, we can factor.
• Option 2: $9h^8-6h^5+4h$ - Not simplified properly.
• Option 3: $9h^7-6h^4+4$ - Only the powers are mismatched.
• Option 4: $9h^7-6h^4$ - Missing a constant.

None of the options provided are equivalent forms.

So, the correct response based on the analysis is that "This rational expression does not have any equivalent forms."

Final Answer: "This rational expression does not have any equivalent forms."