Question
Gina is asked to identify a rational expression. Which of the following could she choose?(1 point)
Responses
9x−48+x√
Start Fraction 9 x minus 4 over 8 plus Start Root x End Root End Fraction
5.32x−1
Start Fraction 5.3 over 2 x minus 1 End Fraction
x32+52x9
Start Fraction x superscript Start Fraction 3 over 2 End Fraction baseline plus 5 over 2 x superscript 9 baseline End Fraction
8x2+70
Responses
9x−48+x√
Start Fraction 9 x minus 4 over 8 plus Start Root x End Root End Fraction
5.32x−1
Start Fraction 5.3 over 2 x minus 1 End Fraction
x32+52x9
Start Fraction x superscript Start Fraction 3 over 2 End Fraction baseline plus 5 over 2 x superscript 9 baseline End Fraction
8x2+70
Answers
GPT-4o mini
A rational expression is a fraction in which both the numerator and the denominator are polynomials. Let's analyze each of the options provided:
1. **\( \frac{9x - 4}{8 + \sqrt{x}} \)**: The numerator \( (9x - 4) \) is a polynomial, but the denominator \( (8 + \sqrt{x}) \) is not a polynomial because of the square root. So this is **not** a rational expression.
2. **\( \frac{5.3}{2x - 1} \)**: The numerator \( (5.3) \) is a constant (which is also a polynomial of degree 0), and the denominator \( (2x - 1) \) is a polynomial. Therefore, this is a rational expression.
3. **\( \frac{x^{\frac{3}{2}} + \frac{5}{2} x^9}{1} \)**: The numerator \( (x^{\frac{3}{2}} + \frac{5}{2} x^9) \) contains \( x^{\frac{3}{2}} \), which is not a polynomial since it has a fractional exponent. So this is **not** a rational expression.
4. **\( 8x^2 + 70 \)**: This is a polynomial but not a rational expression because it is not in the form of a fraction.
Based on this analysis, the only option that qualifies as a rational expression is:
**\( \frac{5.3}{2x - 1} \)**.
1. **\( \frac{9x - 4}{8 + \sqrt{x}} \)**: The numerator \( (9x - 4) \) is a polynomial, but the denominator \( (8 + \sqrt{x}) \) is not a polynomial because of the square root. So this is **not** a rational expression.
2. **\( \frac{5.3}{2x - 1} \)**: The numerator \( (5.3) \) is a constant (which is also a polynomial of degree 0), and the denominator \( (2x - 1) \) is a polynomial. Therefore, this is a rational expression.
3. **\( \frac{x^{\frac{3}{2}} + \frac{5}{2} x^9}{1} \)**: The numerator \( (x^{\frac{3}{2}} + \frac{5}{2} x^9) \) contains \( x^{\frac{3}{2}} \), which is not a polynomial since it has a fractional exponent. So this is **not** a rational expression.
4. **\( 8x^2 + 70 \)**: This is a polynomial but not a rational expression because it is not in the form of a fraction.
Based on this analysis, the only option that qualifies as a rational expression is:
**\( \frac{5.3}{2x - 1} \)**.