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
9h8−6h5+4h9 8th power of h minus 6 5th power of h plus 4 h
9h7−6h49 7th power of h minus 6 4th power of h
This rational expression does not have any equivalent forms
This rational expression does not have any equivalent forms
9h7−6h4+4
Responses
9h8−6h5+4h9 8th power of h minus 6 5th power of h plus 4 h
9h7−6h49 7th power of h minus 6 4th power of h
This rational expression does not have any equivalent forms
This rational expression does not have any equivalent forms
9h7−6h4+4
Answers
GPT-4o mini
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 expression.
1. **Identify the GCF of the coefficients (27, -18, 12)**:
- The GCF of 27, 18, and 12 is 9.
2. **Identify the GCF of the variable parts**:
- The lowest power of \( h \) in the expression is \( h^3 \). Therefore, we can factor out \( h^3 \) along with the coefficients.
3. **Factoring the expression**:
\[
27h^8 - 18h^5 + 12h^3 = 9h^3(3h^5 - 2h^2 + \frac{12}{9}) = 9h^3(3h^5 - 2h^2 + \frac{4}{3})
\]
This expression can be rewritten in a different form, so let's assess the provided options:
- **Option 1**: \( 9h^8 - 6h^5 + 4h \)
- **Option 2**: \( 9h^7 - 6h^4 - 4 \)
- **Option 3**: **This rational expression does not have any equivalent forms**
- **Option 4**: \( 9h^7 - 6h^4 + 4 \)
After comparing against our factored expression, it appears that none of these options accurately represent the correct factored version of the original expression in the correct form.
However, checking for simplifications only from the polynomial \(3h^5 - 2h^2 + \frac{4}{3}\) could lead to further equivalent expressions but doesn't yield any in the present options.
Given the options, the closest understanding would be that **none of the options represent an equivalent form**, hence:
**The correct answer is**: This rational expression does not have any equivalent forms.
1. **Identify the GCF of the coefficients (27, -18, 12)**:
- The GCF of 27, 18, and 12 is 9.
2. **Identify the GCF of the variable parts**:
- The lowest power of \( h \) in the expression is \( h^3 \). Therefore, we can factor out \( h^3 \) along with the coefficients.
3. **Factoring the expression**:
\[
27h^8 - 18h^5 + 12h^3 = 9h^3(3h^5 - 2h^2 + \frac{12}{9}) = 9h^3(3h^5 - 2h^2 + \frac{4}{3})
\]
This expression can be rewritten in a different form, so let's assess the provided options:
- **Option 1**: \( 9h^8 - 6h^5 + 4h \)
- **Option 2**: \( 9h^7 - 6h^4 - 4 \)
- **Option 3**: **This rational expression does not have any equivalent forms**
- **Option 4**: \( 9h^7 - 6h^4 + 4 \)
After comparing against our factored expression, it appears that none of these options accurately represent the correct factored version of the original expression in the correct form.
However, checking for simplifications only from the polynomial \(3h^5 - 2h^2 + \frac{4}{3}\) could lead to further equivalent expressions but doesn't yield any in the present options.
Given the options, the closest understanding would be that **none of the options represent an equivalent form**, hence:
**The correct answer is**: This rational expression does not have any equivalent forms.