Question
Gina is asked to identify a rational expression. Which of the following could she choose?(1 point)
Responses
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
Start Fraction 8 x squared plus 7 over 0 End Fraction
9x−48+x√
Start Fraction 9 x minus 4 over 8 plus Start Root x End Root End Fraction
5.32x−1
Responses
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
Start Fraction 8 x squared plus 7 over 0 End Fraction
9x−48+x√
Start Fraction 9 x minus 4 over 8 plus Start Root x End Root End Fraction
5.32x−1
Answers
GPT-4o mini
A rational expression is defined as the quotient of two polynomials. Among the options provided, let’s analyze each one:
1. **\( \frac{x^{\frac{3}{2}} + \frac{5}{2} x^9}{1} \)** - This expression is not a rational expression because \(x^{\frac{3}{2}}\) involves a fractional exponent, which is not a polynomial.
2. **\( \frac{8x^2 + 7}{0} \)** - This expression is problematic because division by zero is undefined. Although the numerator is a polynomial, the denominator being zero disqualifies it as a rational expression.
3. **\( \frac{9x - 4}{8 + \sqrt{x}} \)** - The denominator includes \(\sqrt{x}\), which makes it not a polynomial. As such, this is also not a rational expression.
4. **\( 5.32x^{-1} \)** - This expression can be rewritten as \( \frac{5.32}{x} \), which is the ratio of two polynomials (5.32 is considered a polynomial of degree zero, and \(x\) is a polynomial of degree one). Thus, this qualifies as a rational expression.
Based on this analysis, the correct answer is **\( 5.32x^{-1} \)**.
1. **\( \frac{x^{\frac{3}{2}} + \frac{5}{2} x^9}{1} \)** - This expression is not a rational expression because \(x^{\frac{3}{2}}\) involves a fractional exponent, which is not a polynomial.
2. **\( \frac{8x^2 + 7}{0} \)** - This expression is problematic because division by zero is undefined. Although the numerator is a polynomial, the denominator being zero disqualifies it as a rational expression.
3. **\( \frac{9x - 4}{8 + \sqrt{x}} \)** - The denominator includes \(\sqrt{x}\), which makes it not a polynomial. As such, this is also not a rational expression.
4. **\( 5.32x^{-1} \)** - This expression can be rewritten as \( \frac{5.32}{x} \), which is the ratio of two polynomials (5.32 is considered a polynomial of degree zero, and \(x\) is a polynomial of degree one). Thus, this qualifies as a rational expression.
Based on this analysis, the correct answer is **\( 5.32x^{-1} \)**.