Question

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)
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 \(27h^8 - 18h^5 + 12h^3h\), we first need to rewrite the last term and then factor out the greatest common factor (GCF).

1. Rewrite the last term:
\[
12h^3h = 12h^{3+1} = 12h^4
\]
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."**